diff --git a/README.md b/README.md index b83a2ce..414b65b 100644 --- a/README.md +++ b/README.md @@ -7,7 +7,7 @@ This is a list of my publications. I have included LaTeX code for preprint versi
  1. With Laurent Marcoux and Yuanhang Zhang, Kaplansky's problem and unitary orbits in matrix amplifications. https://arxiv.org/abs/2508.13834
  2. Universal covering groups of unitary groups of von Neumann algebras. Studia Mathematica (2025). doi:10.4064/sm240829-14-1 (arxiv preprint: 2408.13710)
    3. -
  3. With Aaron Tikuisis, *Polar decomposition in algebraic K-theory. Journal of Operator Theory (2025). doi:10.7900/jot.2023may29.2436 (arxiv preprint: 2303.16248)
    4. +
  4. With Aaron Tikuisis, Polar decomposition in algebraic K-theory. Journal of Operator Theory (2025). doi:10.7900/jot.2023may29.2436 (arxiv preprint: 2303.16248)
  5. Tensorially absorbing inclusions of C*-algebras. Canadian Journal of Mathematics (2024). doi:10.4153/S0008414X24000324 (arxiv preprint: 2211.14974)
  6. Unitary groups, K-theory and traces. Glasgow Mathematical Journal (2023). doi:10.1017/S0017089523000447 (arxiv preprint: 2305.15989)
diff --git a/Tensorially absorbing inclusions/biblio.bib b/Tensorially absorbing inclusions/biblio.bib new file mode 100644 index 0000000..0f1f425 --- /dev/null +++ b/Tensorially absorbing inclusions/biblio.bib @@ -0,0 +1,2805 @@ +#### AAAAAA + +# Akemann-Anderson-Pedersen +@article {AkemannAndersonPedersen86, + AUTHOR = {Akemann, Charles A. and Anderson, Joel and Pedersen, Gert K.}, + TITLE = {Excising states of {C}*-algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {38}, + YEAR = {1986}, + NUMBER = {5}, + PAGES = {1239--1260}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (46L30)}, + MRNUMBER = {869724}, +MRREVIEWER = {J. W. Bunce}, + DOI = {10.4153/CJM-1986-063-7}, + URL = {https://doi.org/10.4153/CJM-1986-063-7}, +} + +# Al-Rawashdeh-Booth-Giordano +@article {AlBoothGiordano12, + AUTHOR = {Al-Rawashdeh, Ahmed and Booth, Andrew and Giordano, Thierry}, + TITLE = {Unitary groups as a complete invariant}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {262}, + YEAR = {2012}, + NUMBER = {11}, + PAGES = {4711--4730}, + ISSN = {0022-1236}, + MRCLASS = {46L35}, + MRNUMBER = {2913684}, + DOI = {10.1016/j.jfa.2012.03.016}, + URL = {https://doi.org/10.1016/j.jfa.2012.03.016}, +} + +# Amini-Golestani-Jamali-Phillips +@article{AGJP22, + title={Group actions on simple tracially $\mathcal{Z}$-absorbing {C}*-algebras}, + author={Amini, Massoud and Golestani, Nasser and Jamali, Saeid and Phillips, N. Christopher}, + journal={arXiv preprint arXiv:2204.03615}, + year={2022}, +} + +# Amrutam-Kalantar +@article {AmrutamKalantar20, + AUTHOR = {Amrutam, Tattwamasi and Kalantar, Mehrdad}, + TITLE = {On simplicity of intermediate {C}*-algebras}, + JOURNAL = {Ergodic Theory Dynam. Systems}, + FJOURNAL = {Ergodic Theory and Dynamical Systems}, + VOLUME = {40}, + YEAR = {2020}, + NUMBER = {12}, + PAGES = {3181--3187}, + ISSN = {0143-3857}, + MRCLASS = {46L55 (37A55 46L10 46L35 46L89)}, + MRNUMBER = {4170599}, +MRREVIEWER = {Bruno Brogni Uggioni}, + DOI = {10.1017/etds.2019.34}, + URL = {https://doi.org/10.1017/etds.2019.34}, +} + +# Ando-Haagerup +@article {AndoHaagerup14, + AUTHOR = {Ando, Hiroshi and Haagerup, Uffe}, + TITLE = {Ultraproducts of von {N}eumann algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {266}, + YEAR = {2014}, + NUMBER = {12}, + PAGES = {6842--6913}, + ISSN = {0022-1236}, + MRCLASS = {46M07 (46L10)}, + MRNUMBER = {3198856}, +MRREVIEWER = {E. St{\o}rmer}, + DOI = {10.1016/j.jfa.2014.03.013}, + URL = {https://doi.org/10.1016/j.jfa.2014.03.013}, +} + +# Ara-Mathieu +@book {AraMathieubook, + AUTHOR = {Ara, Pere and Mathieu, Martin}, + TITLE = {Local multipliers of {C}*-algebras}, + SERIES = {Springer Monographs in Mathematics}, + PUBLISHER = {Springer-Verlag London, Ltd., London}, + YEAR = {2003}, + PAGES = {xii+319}, + ISBN = {1-85233-237-9}, + MRCLASS = {46L05 (16S90 46Kxx 46L40 47B47 47L25)}, + MRNUMBER = {1940428}, +MRREVIEWER = {Michael Frank}, + DOI = {10.1007/978-1-4471-0045-4}, + URL = {https://doi.org/10.1007/978-1-4471-0045-4}, +} + +#### BBBBBBBBBBBBBBBBBBBBBBBB + +#Bresar +@article {Bresar93, + AUTHOR = {Bre\v{s}ar, Matej}, + TITLE = {Commuting traces of biadditive mappings, + commutativity-preserving mappings and {L}ie mappings}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {335}, + YEAR = {1993}, + NUMBER = {2}, + PAGES = {525--546}, + ISSN = {0002-9947}, + MRCLASS = {16W25 (16N60 16W10)}, + MRNUMBER = {1069746}, +MRREVIEWER = {Howard E. Bell}, + DOI = {10.2307/2154392}, + URL = {https://doi.org/10.2307/2154392}, +} + +#Bisch +@article {Bisch90, + AUTHOR = {Bisch, Dietmar H.}, + TITLE = {On the existence of central sequences in subfactors}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {321}, + YEAR = {1990}, + NUMBER = {1}, + PAGES = {117--128}, + ISSN = {0002-9947}, + MRCLASS = {46L35}, + MRNUMBER = {1005075}, +MRREVIEWER = {V. S. Sunder}, + DOI = {10.2307/2001593}, + URL = {https://doi.org/10.2307/2001593}, +} + +@article {Bisch94, + AUTHOR = {Bisch, Dietmar H.}, + TITLE = {Central sequences in subfactors. {II}}, + JOURNAL = {Proc. Amer. Math. Soc.}, + FJOURNAL = {Proceedings of the American Mathematical Society}, + VOLUME = {121}, + YEAR = {1994}, + NUMBER = {3}, + PAGES = {725--731}, + ISSN = {0002-9939}, + MRCLASS = {46L37}, + MRNUMBER = {1209417}, +MRREVIEWER = {V. S. Sunder}, + DOI = {10.2307/2160268}, + URL = {https://doi.org/10.2307/2160268}, +} + +# Blackadar +@article {Blackadar90, + AUTHOR = {Blackadar, Bruce}, + TITLE = {Symmetries of the {CAR} algebra}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {131}, + YEAR = {1990}, + NUMBER = {3}, + PAGES = {589--623}, + ISSN = {0003-486X}, + MRCLASS = {46L80 (19K14)}, + MRNUMBER = {1053492}, +MRREVIEWER = {Bola O. Balogun}, + DOI = {10.2307/1971472}, + URL = {https://doi.org/10.2307/1971472}, +} + +@book {BlackadarBook, + AUTHOR = {Blackadar, Bruce}, + TITLE = {Operator algebras}, + SERIES = {Encyclopaedia of Mathematical Sciences}, + VOLUME = {122}, + NOTE = {Theory of {C}*-algebras and von {N}eumann algebras, + Operator Algebras and Non-commutative Geometry, {III}}, + PUBLISHER = {Springer-Verlag, Berlin}, + YEAR = {2006}, + PAGES = {xx+517}, + ISBN = {978-3-540-28486-4; 3-540-28486-9}, + MRCLASS = {46L05 (46L10 46L80)}, + MRNUMBER = {2188261}, +MRREVIEWER = {Paul Jolissaint}, + DOI = {10.1007/3-540-28517-2}, + URL = {https://doi.org/10.1007/3-540-28517-2}, +} + +# Blacakdar-Kumjian-Rordam +@article {BKR92, + AUTHOR = {Blackadar, Bruce and Kumjian, Alexander and R{\o}rdam, Mikael}, + TITLE = {Approximately central matrix units and the structure of + noncommutative tori}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {6}, + YEAR = {1992}, + NUMBER = {3}, + PAGES = {267--284}, + ISSN = {0920-3036}, + MRCLASS = {46L87 (46L05)}, + MRNUMBER = {1189278}, +MRREVIEWER = {Chi Wai Leung}, + DOI = {10.1007/BF00961466}, + URL = {https://doi.org/10.1007/BF00961466}, +} + +# Booth +@mastersthesis{Booth98, + title={The unitary group as a complete invariant for simple unital {AF} algebras}, + author={Booth, Andrew}, + year={1998}, + school={University of Ottawa (Canada)}, +} + +# Bosa-Gabe-Sims-White +@article {BGSW22, + AUTHOR = {Bosa, Joan and Gabe, James and Sims, Aidan and White, Stuart}, + TITLE = {The nuclear dimension of $\mathcal{O}_\infty$-stable {C}*-algebras}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {401}, + YEAR = {2022}, + PAGES = {Paper No. 108250, 51}, + ISSN = {0001-8708}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {4392219}, + DOI = {10.1016/j.aim.2022.108250}, + URL = {https://doi.org/10.1016/j.aim.2022.108250}, +} + +# Brattelo-Stormer-Kishimoto-Rordam +@article {BSKR93, + AUTHOR = {Bratteli, Ola and St{\o}rmer, Erling and Kishimoto, Akitaka and + R{\o}rdam, Mikael}, + TITLE = {The crossed product of a {UHF} algebra by a shift}, + JOURNAL = {Ergodic Theory Dynam. Systems}, + FJOURNAL = {Ergodic Theory and Dynamical Systems}, + VOLUME = {13}, + YEAR = {1993}, + NUMBER = {4}, + PAGES = {615--626}, + ISSN = {0143-3857}, + MRCLASS = {46L55 (46L35)}, + MRNUMBER = {1257025}, +MRREVIEWER = {Robert J. Archbold}, + DOI = {10.1017/S0143385700007574}, + URL = {https://doi.org/10.1017/S0143385700007574}, +} + +# Brown-Ozawa +@book {BrownOzawa, + AUTHOR = {Brown, Nathanial P. and Ozawa, Narutaka}, + TITLE = {{C}*-algebras and finite-dimensional approximations}, + SERIES = {Graduate Studies in Mathematics}, + VOLUME = {88}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {2008}, + PAGES = {xvi+509}, + ISBN = {978-0-8218-4381-9; 0-8218-4381-8}, + MRCLASS = {46L05 (43A07 46-02 46L10)}, + MRNUMBER = {2391387}, +MRREVIEWER = {Mikael R{\o}rdam}, + DOI = {10.1090/gsm/088}, + URL = {https://doi.org/10.1090/gsm/088}, +} + +# Burnstein +@article {Burstein10, + AUTHOR = {Burstein, Richard D.}, + TITLE = {Commuting square subfactors and central sequences}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {21}, + YEAR = {2010}, + NUMBER = {1}, + PAGES = {117--131}, + ISSN = {0129-167X}, + MRCLASS = {46L37}, + MRNUMBER = {2642989}, +MRREVIEWER = {Toshihiko Masuda}, + DOI = {10.1142/S0129167X10005945}, + URL = {https://doi.org/10.1142/S0129167X10005945}, +} + +### CCCCCCCCCCCCCCCCCCCCCC + +# Cameron-Smith +@article {CameronSmith19, + AUTHOR = {Cameron, Jan and Smith, Roger R.}, + TITLE = {A {G}alois correspondence for reduced crossed products of + simple {C}*-algebras by discrete groups}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {71}, + YEAR = {2019}, + NUMBER = {5}, + PAGES = {1103--1125}, + ISSN = {0008-414X}, + MRCLASS = {46L55 (46L40)}, + MRNUMBER = {4010423}, +MRREVIEWER = {Robert S. Doran}, + DOI = {10.4153/cjm-2018-014-6}, + URL = {https://doi.org/10.4153/cjm-2018-014-6}, +} + +#Carrion-Gabe-Schafhauser-Tikuisis-White +@misc{CGSTW, + title={Classifying *-homomorphisms {I}: simple nuclear {C}*-algebras}, + author={Carri{\'o}n, Jos{\'e} R. and Gabe, James and Schafhauser, Christopher and Tikuisis, Aaron, and White, Stuart}, + note={preprint}, +} + +# Castillejos-Evington-Tikuisis-White-Winter +@article {CETWW21, + AUTHOR = {Castillejos, Jorge and Evington, Samuel and Tikuisis, Aaron + and White, Stuart and Winter, Wilhelm}, + TITLE = {Nuclear dimension of simple {C}*-algebras}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {224}, + YEAR = {2021}, + NUMBER = {1}, + PAGES = {245--290}, + ISSN = {0020-9910}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {4228503}, +MRREVIEWER = {Changguo Wei}, + DOI = {10.1007/s00222-020-01013-1}, + URL = {https://doi.org/10.1007/s00222-020-01013-1}, +} + +# Castillejos-Evington-Tikuisis-White +@article{CETW19, + title={Uniform property {G}amma}, + author={Castillejos, Jorge and Evington, Samuel and Tikuisis, Aaron and White, Stuart}, + journal={arXiv preprint arXiv:1912.04207}, + year={2019} +} + +@article {CETW22, + AUTHOR = {Castillejos, Jorge and Evington, Samuel and Tikuisis, Aaron + and White, Stuart}, + TITLE = {Uniform property {$\Gamma$}}, + JOURNAL = {Int. Math. Res. Not. IMRN}, + FJOURNAL = {International Mathematics Research Notices. IMRN}, + YEAR = {2022}, + NUMBER = {13}, + PAGES = {9864--9908}, + ISSN = {1073-7928}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {4447140}, + DOI = {10.1093/imrn/rnaa282}, + URL = {https://doi.org/10.1093/imrn/rnaa282}, +} + +# Chand-Robert +@article {ChandRobert23, + AUTHOR = {Chand, Abhinav and Robert, Leonel}, + TITLE = {Simplicity, bounded normal generation, and automatic + continuity of groups of unitaries}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {415}, + YEAR = {2023}, + PAGES = {Paper No. 108894, 52}, + ISSN = {0001-8708}, + MRCLASS = {46L05 (22E65 46H40)}, + MRNUMBER = {4543451}, + DOI = {10.1016/j.aim.2023.108894}, + URL = {https://doi.org/10.1016/j.aim.2023.108894}, +} + +# Choi-Effros +@article {ChoiEffros76, + AUTHOR = {Choi, Man Duen and Effros, Edward G.}, + TITLE = {The completely positive lifting problem for + {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {104}, + YEAR = {1976}, + NUMBER = {3}, + PAGES = {585--609}, + ISSN = {0003-486X}, + MRCLASS = {46L05}, + MRNUMBER = {417795}, +MRREVIEWER = {Maurice J. Dupr\'{e}}, + DOI = {10.2307/1970968}, + URL = {https://doi.org/10.2307/1970968}, +} + +# Connes +@article {Connes75, + AUTHOR = {Connes, Alain}, + TITLE = {Classification of automorphisms of hyperfinite factors of type {II}$_1$ and {II}$_{\infty}$ and application to type {III} factors}, + JOURNAL = {Bull. Amer. Math. Soc.}, + FJOURNAL = {Bulletin of the American Mathematical Society}, + VOLUME = {81}, + YEAR = {1975}, + NUMBER = {6}, + PAGES = {1090--1092}, + ISSN = {0002-9904}, + MRCLASS = {46L10}, + MRNUMBER = {388117}, +MRREVIEWER = {Hisashi Choda}, + DOI = {10.1090/S0002-9904-1975-13929-2}, + URL = {https://doi.org/10.1090/S0002-9904-1975-13929-2}, +} + +@article {Connes75b, + AUTHOR = {Connes, Alain}, + TITLE = {A factor not anti-isomorphic to itself}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {101}, + YEAR = {1975}, + PAGES = {536--554}, + ISSN = {0003-486X}, + MRCLASS = {46L10}, + MRNUMBER = {370209}, +MRREVIEWER = {E. St{\o}rmer}, + DOI = {10.2307/1970940}, + URL = {https://doi.org/10.2307/1970940}, +} + +@article {Connes76, + AUTHOR = {Connes, Alain}, + TITLE = {Classification of injective factors Cases {II}$_1$, {II}$_\infty$, {III}$_\lambda$, $\lambda \neq 1$}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {104}, + YEAR = {1976}, + NUMBER = {1}, + PAGES = {73--115}, + ISSN = {0003-486X}, + MRCLASS = {46L10}, + MRNUMBER = {454659}, +MRREVIEWER = {Fran\c{c}ois Combes}, + DOI = {10.2307/1971057}, + URL = {https://doi.org/10.2307/1971057}, +} + + +@article {Connes77, + AUTHOR = {Connes, Alain}, + TITLE = {Periodic automorphisms of the hyperfinite factor of type {II}$_1$}, + JOURNAL = {Acta Sci. Math. (Szeged)}, + FJOURNAL = {Acta Universitatis Szegediensis. Acta Scientiarum + Mathematicarum}, + VOLUME = {39}, + YEAR = {1977}, + NUMBER = {1-2}, + PAGES = {39--66}, + ISSN = {0001-6969}, + MRCLASS = {46L10}, + MRNUMBER = {448101}, +MRREVIEWER = {Yoshinori Haga}, +} + +@article {Connes85, + AUTHOR = {Connes, Alain}, + TITLE = {FACTORS OF TYPE {III}$_1$, PROPERTY {L}$_\lambda'$ AND CLOSURE OF INNER AUTOMORPHISMS}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {14}, + YEAR = {1985}, + NUMBER = {1}, + PAGES = {189--211}, + ISSN = {0379-4024}, + MRCLASS = {46L35}, + MRNUMBER = {789385}, +MRREVIEWER = {M. Takesaki}, +} + +# Conway +@book {Conway19, + AUTHOR = {Conway, John B.}, + TITLE = {A course in functional analysis}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {96}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1985}, + PAGES = {xiv+404}, + ISBN = {0-387-96042-2}, + MRCLASS = {46-01 (47-01)}, + MRNUMBER = {768926}, +MRREVIEWER = {Greg Robel}, + DOI = {10.1007/978-1-4757-3828-5}, + URL = {https://doi.org/10.1007/978-1-4757-3828-5}, +} + +# Cuntz +@article {Cuntz77, + AUTHOR = {Cuntz, Joachim}, + TITLE = {Simple {C}*-algebras generated by isometries}, + JOURNAL = {Comm. Math. Phys.}, + FJOURNAL = {Communications in Mathematical Physics}, + VOLUME = {57}, + YEAR = {1977}, + NUMBER = {2}, + PAGES = {173--185}, + ISSN = {0010-3616}, + MRCLASS = {46L05}, + MRNUMBER = {467330}, +MRREVIEWER = {E. St{\o}rmer}, + URL = {http://projecteuclid.org/euclid.cmp/1103901288}, +} + +@article {Cuntz81, + AUTHOR = {Cuntz, Joachim}, + TITLE = {{K}-theory for certain {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {113}, + YEAR = {1981}, + NUMBER = {1}, + PAGES = {181--197}, + ISSN = {0003-486X}, + MRCLASS = {46L05 (16A54 18G99 46M20 58G12)}, + MRNUMBER = {604046}, +MRREVIEWER = {Vern Paulsen}, + DOI = {10.2307/1971137}, + URL = {https://doi.org/10.2307/1971137}, +} + +# Cuntz-Pedersen +@article{CuntzPedersen79, + title={Equivalence and traces on {C}*-algebras}, + author={Cuntz, Joachim and Pedersen, Gert K.}, + journal={Journal of Functional Analysis}, + volume={33}, + number={2}, + pages={135--164}, + year={1979}, + publisher={Elsevier} +} +@article {CuntzPedersen79, + AUTHOR = {Cuntz, Joachim and Pedersen, Gert Kjaerg\.{a}rd}, + TITLE = {Equivalence and traces on {C}*-algebras}, + JOURNAL = {J. Functional Analysis}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {33}, + YEAR = {1979}, + NUMBER = {2}, + PAGES = {135--164}, + ISSN = {0022-1236}, + MRCLASS = {46L05}, + MRNUMBER = {546503}, +MRREVIEWER = {Richard I. Loebl}, + DOI = {10.1016/0022-1236(79)90108-3}, + URL = {https://doi.org/10.1016/0022-1236(79)90108-3}, +} + +### DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD +# Dadarlat +@article {Dadarlat95, + AUTHOR = {Dadarlat, Marius}, + TITLE = {Reduction to dimension three of local spectra of real rank + zero {C}*-algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {460}, + YEAR = {1995}, + PAGES = {189--212}, + ISSN = {0075-4102}, + MRCLASS = {46L85 (19K14 46L35)}, + MRNUMBER = {1316577}, +MRREVIEWER = {Terry A. Loring}, + DOI = {10.1515/crll.1995.460.189}, + URL = {https://doi.org/10.1515/crll.1995.460.189}, +} + +# Dadarlat-Loring +@article {MR1404333, + AUTHOR = {Dadarlat, Marius and Loring, Terry A.}, + TITLE = {A universal multicoefficient theorem for the {K}asparov + groups}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {84}, + YEAR = {1996}, + NUMBER = {2}, + PAGES = {355--377}, + ISSN = {0012-7094}, + MRCLASS = {46L80 (19K35 46L35)}, + MRNUMBER = {1404333}, +MRREVIEWER = {Kevin McClanahan}, + DOI = {10.1215/S0012-7094-96-08412-4}, + URL = {https://doi.org/10.1215/S0012-7094-96-08412-4}, +} + +@article {DadarlatWinter09, + AUTHOR = {Dadarlat, Marius and Winter, Wilhelm}, + TITLE = {On the {KK}-theory of strongly self-absorbing + {C}*-algebras}, + JOURNAL = {Math. Scand.}, + FJOURNAL = {Mathematica Scandinavica}, + VOLUME = {104}, + YEAR = {2009}, + NUMBER = {1}, + PAGES = {95--107}, + ISSN = {0025-5521}, + MRCLASS = {46L80 (19K35 46L05)}, + MRNUMBER = {2498373}, +MRREVIEWER = {Vladimir Manuilov}, + DOI = {10.7146/math.scand.a-15086}, + URL = {https://doi.org/10.7146/math.scand.a-15086}, +} + +# Davidson +@book {DavidsonBook1, + AUTHOR = {Davidson, Kenneth R.}, + TITLE = {{C}*-algebras by example}, + SERIES = {Fields Institute Monographs}, + VOLUME = {6}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {1996}, + PAGES = {xiv+309}, + ISBN = {0-8218-0599-1}, + MRCLASS = {46Lxx (46-01)}, + MRNUMBER = {1402012}, +MRREVIEWER = {Robert S. Doran}, + DOI = {10.1090/fim/006}, + URL = {https://doi.org/10.1090/fim/006}, +} + +#de Lacerda Mortari +@book {deLacerdaMortari09, + AUTHOR = {Mortari, Fernando de Lacerda}, + TITLE = {Tracial state spaces of higher stable rank simple + {C}*-algebras}, + NOTE = {Thesis (Ph.D.)--University of Toronto (Canada)}, + PUBLISHER = {ProQuest LLC, Ann Arbor, MI}, + YEAR = {2009}, + PAGES = {41}, + ISBN = {978-0494-61035-0}, + MRCLASS = {Thesis}, + MRNUMBER = {2753146}, + URL = + {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:NR61035}, +} + +# de la Harpe +@article {dlHarpe13, + AUTHOR = {de la Harpe, Pierre}, + TITLE = {Fuglede--{K}adison determinant: theme and variations}, + JOURNAL = {Proc. Natl. Acad. Sci. USA}, + FJOURNAL = {Proceedings of the National Academy of Sciences of the United + States of America}, + VOLUME = {110}, + YEAR = {2013}, + NUMBER = {40}, + PAGES = {15864--15877}, + ISSN = {0027-8424}, + MRCLASS = {46L10}, + MRNUMBER = {3363445}, + DOI = {10.1073/pnas.1202059110}, + URL = {https://doi.org/10.1073/pnas.1202059110}, +} + +# de la Harpe-Skandalis +@article {dlHS84a, + AUTHOR = {de la Harpe, Pierre and Skandalis, Georges}, + TITLE = {D\'{e}terminant associ\'{e} \`a une trace sur une alg\'{e}bre de {B}anach}, + JOURNAL = {Ann. Inst. Fourier (Grenoble)}, + FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier}, + VOLUME = {34}, + YEAR = {1984}, + NUMBER = {1}, + PAGES = {241--260}, + ISSN = {0373-0956}, + MRCLASS = {46L80 (18F25 19K14 19K56 46L05 58G12)}, + MRNUMBER = {743629}, +MRREVIEWER = {G. A. Elliott}, + URL = {http://www.numdam.org/item?id=AIF_1984__34_1_241_0}, +} + +@article {dlHS84b, + AUTHOR = {de la Harpe, Pierre and Skandalis, Georges}, + TITLE = {Produits finis de commutateurs dans les {C}*-alg\`ebres}, + JOURNAL = {Ann. Inst. Fourier (Grenoble)}, + FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier}, + VOLUME = {34}, + YEAR = {1984}, + NUMBER = {4}, + PAGES = {169--202}, + ISSN = {0373-0956}, + MRCLASS = {46L05 (18F25 19B99 19C09 19K14 46L80 58G12)}, + MRNUMBER = {766279}, +MRREVIEWER = {G. A. Elliott}, + URL = {http://www.numdam.org/item?id=AIF_1984__34_4_169_0}, +} + +# Dummit-Foote +@book {DummitFoote, + AUTHOR = {Dummit, David S. and Foote, Richard M.}, + TITLE = {Abstract algebra}, + EDITION = {Third}, + PUBLISHER = {John Wiley \& Sons, Inc., Hoboken, NJ}, + YEAR = {2004}, + PAGES = {xii+932}, + ISBN = {0-471-43334-9}, + MRCLASS = {00-01 (16-01 20-01)}, + MRNUMBER = {2286236}, +} + +# Dye +@article {Dye53, + AUTHOR = {Dye, H. A.}, + TITLE = {The unitary structure in finite rings of operators}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {20}, + YEAR = {1953}, + PAGES = {55--69}, + ISSN = {0012-7094}, + MRCLASS = {46.3X}, + MRNUMBER = {52695}, +MRREVIEWER = {F. I. Mautner}, + URL = {http://projecteuclid.org/euclid.dmj/1077465064}, +} + +@article {Dye55, + AUTHOR = {Dye, H. A.}, + TITLE = {On the geometry of projections in certain operator algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {61}, + YEAR = {1955}, + PAGES = {73--89}, + ISSN = {0003-486X}, + MRCLASS = {46.0X}, + MRNUMBER = {66568}, +MRREVIEWER = {J. Dixmier}, + DOI = {10.2307/1969620}, + URL = {https://doi.org/10.2307/1969620}, +} + +### EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE +# Elliott +@article {Elliott76, + AUTHOR = {Elliott, George A.}, + TITLE = {On the classification of inductive limits of sequences of + semisimple finite-dimensional algebras}, + JOURNAL = {J. Algebra}, + FJOURNAL = {Journal of Algebra}, + VOLUME = {38}, + YEAR = {1976}, + NUMBER = {1}, + PAGES = {29--44}, + ISSN = {0021-8693}, + MRCLASS = {46L05 (16A46)}, + MRNUMBER = {397420}, +MRREVIEWER = {Horst Behncke}, + DOI = {10.1016/0021-8693(76)90242-8}, + URL = {https://doi.org/10.1016/0021-8693(76)90242-8}, +} + +@article {Elliott93, + AUTHOR = {Elliott, George A.}, + TITLE = {On the classification of {C}*-algebras of real rank zero}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {443}, + YEAR = {1993}, + PAGES = {179--219}, + ISSN = {0075-4102}, + MRCLASS = {46L05 (19K14 46L80)}, + MRNUMBER = {1241132}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1515/crll.1993.443.179}, + URL = {https://doi.org/10.1515/crll.1993.443.179}, +} + +@article{Elliott22, +title="K-Theory and Traces", +author="Elliott, George A.", +journal="C. R. Math. Rep. Acad. Sci. Canada", +volume="44", +number="1", +pages="1--15", +year="2022" +} + +# Elliott-Gong-Lin-Niu +@article{EGLN15, + title={On the classification of simple amenable {C}*-algebras with finite decomposition rank, {II}}, + author={Elliott, George A. and Gong, Guihua and Lin, H. and Niu, Zhuang}, + journal={arXiv preprint arXiv:1507.03437}, + year={2015} +} + +# Elliott-Li-Niu +@article{ElliottLiNiu22, + title={A remark on {V}illadsen algebras}, + author={Elliott, George A. and Li, Chun G. and Niu, Zhuang}, + journal={arXiv preprint arXiv:2209.10649}, + year={2022} +} + +# Elliott-Niu +@article {ElliottNiu12, + AUTHOR = {Elliott, George A. and Niu, Zhuang}, + TITLE = {Extended rotation algebras: adjoining spectral projections to + rotation algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {665}, + YEAR = {2012}, + PAGES = {1--71}, + ISSN = {0075-4102}, + MRCLASS = {46L05}, + MRNUMBER = {2908740}, +MRREVIEWER = {William Paschke}, + DOI = {10.1515/CRELLE.2011.112}, + URL = {https://doi.org/10.1515/CRELLE.2011.112}, +} + +@article {ElliottNiu15, + AUTHOR = {Elliott, George A. and Niu, Zhuang}, + TITLE = {All irrational extended rotation algebras are {AF} algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {67}, + YEAR = {2015}, + NUMBER = {4}, + PAGES = {810--826}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {3361014}, +MRREVIEWER = {Katsutoshi Kawashima}, + DOI = {10.4153/CJM-2014-022-5}, + URL = {https://doi.org/10.4153/CJM-2014-022-5}, +} + +@incollection {ElliottNiu16, + AUTHOR = {Elliott, George A. and Niu, Zhuang}, + TITLE = {On the classification of simple amenable {C}*-algebras with + finite decomposition rank}, + BOOKTITLE = {Operator algebras and their applications}, + SERIES = {Contemp. Math.}, + VOLUME = {671}, + PAGES = {117--125}, + PUBLISHER = {Amer. Math. Soc., Providence, RI}, + YEAR = {2016}, + MRCLASS = {46L35}, + MRNUMBER = {3546681}, + DOI = {10.1090/conm/671/13506}, + URL = {https://doi.org/10.1090/conm/671/13506}, +} + +# Elliott-Villadsen +@article {ElliottVilladsen00, + AUTHOR = {Elliott, George A. and Villadsen, Jesper}, + TITLE = {Perforated ordered {$K_0$}-groups}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {52}, + YEAR = {2000}, + NUMBER = {6}, + PAGES = {1164--1191}, + ISSN = {0008-414X}, + MRCLASS = {46L80 (19K14 46L05)}, + MRNUMBER = {1794301}, + DOI = {10.4153/CJM-2000-049-9}, + URL = {https://doi.org/10.4153/CJM-2000-049-9}, +} + +# Echterhoff-Rordam +@article{EchterhoffRordam21, + title={Inclusions of {C}*-algebras arising from fixed-point algebras}, + author={Echterhoff, Siegfried and R{\o}rdam, Mikael}, + journal={arXiv preprint arXiv:2108.08832}, + year={2021} +} + +# Enders-Schemaitat-Tikuisis +@article{EndersSchemaitatTikuisis23, + title={Corrigendum to ``{$K$}-theoretic characterization of {C}*-algebras with approximately inner flip''}, + author={Enders, Dominic and Schemaitat, Andr{\'e} and Tikuisis, Aaron}, + journal={arXiv preprint arXiv:2303.11106}, + year={2023} +} + +### FFFFFFFFFFFFFFFFFFFFFF + +#FangGeLi +@article {FangGeLi06, + AUTHOR = {Fang, Junsheng and Ge, Liming and Li, Weihua}, + TITLE = {Central sequence algebras of von {N}eumann algebras}, + JOURNAL = {Taiwanese J. Math.}, + FJOURNAL = {Taiwanese Journal of Mathematics}, + VOLUME = {10}, + YEAR = {2006}, + NUMBER = {1}, + PAGES = {187--200}, + ISSN = {1027-5487}, + MRCLASS = {46L10 (46M07)}, + MRNUMBER = {2186173}, +MRREVIEWER = {H. Halpern}, + DOI = {10.11650/twjm/1500403810}, + URL = {https://doi.org/10.11650/twjm/1500403810}, +} + +# Farah-Hirshberg +@article{FarahHirshberg16, + title={A simple {AF} algebra not isomorphic to its opposite}, + author={Farah, Ilijas and Hirshberg, Ilan}, + journal={arXiv preprint arXiv:1612.01170}, + year={2016} +} + +@article {FarahHirshberg17, + AUTHOR = {Farah, Ilijas and Hirshberg, Ilan}, + TITLE = {Simple nuclear {C}*-algebras not isomorphic to their + opposites}, + JOURNAL = {Proc. Natl. Acad. Sci. USA}, + FJOURNAL = {Proceedings of the National Academy of Sciences of the United + States of America}, + VOLUME = {114}, + YEAR = {2017}, + NUMBER = {24}, + PAGES = {6244--6249}, + ISSN = {0027-8424}, + MRCLASS = {46L10}, + MRNUMBER = {3667529}, +MRREVIEWER = {Fyodor A. Sukochev}, + DOI = {10.1073/pnas.1619936114}, + URL = {https://doi.org/10.1073/pnas.1619936114}, +} + +# Folland +@book {Follandbook16, + AUTHOR = {Folland, Gerald B.}, + TITLE = {A course in abstract harmonic analysis}, + SERIES = {Textbooks in Mathematics}, + EDITION = {Second}, + PUBLISHER = {CRC Press, Boca Raton, FL}, + YEAR = {2016}, + PAGES = {xiii+305 pp.+loose errata}, + ISBN = {978-1-4987-2713-6}, + MRCLASS = {43-01 (22-01 42-01 46-01)}, + MRNUMBER = {3444405}, +MRREVIEWER = {D. L. Salinger}, +} + +### GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG +# Gabe +@article{Gabe19, + title={Classification of $\mathcal{O}_\infty$-stable {C}*-algebras}, + author={Gabe, James}, + journal={arXiv preprint arXiv:1910.06504}, + year={2019} +} + +@article {Gabe20, + AUTHOR = {Gabe, James}, + TITLE = {A new proof of {K}irchberg's $\mathcal{O}_2$-stable classification}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {761}, + YEAR = {2020}, + PAGES = {247--289}, + ISSN = {0075-4102}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {4080250}, +MRREVIEWER = {Francesc Perera}, + DOI = {10.1515/crelle-2018-0010}, + URL = {https://doi.org/10.1515/crelle-2018-0010}, +} + +# Gardella +@article{Gardella17, + title={{R}okhlin-type properties for group actions on {C}*-algebras}, + author={Gardella, Eusebio}, + journal={Lecture Notes, IPM, Tehran}, + year={2017} +} + +@book {GardellaBook, + AUTHOR = {Gardella, Emilio Eusebio}, + TITLE = {Compact group actions on {C}*-algebras: classification, + non-classifiability, and crossed products and rigidity results + for {L}p-operator algebras}, + NOTE = {Thesis (Ph.D.)--University of Oregon}, + PUBLISHER = {ProQuest LLC, Ann Arbor, MI}, + YEAR = {2015}, + PAGES = {713}, + ISBN = {978-1321-96796-8}, + MRCLASS = {Thesis}, + MRNUMBER = {3407494}, + URL = + {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3717348}, +} + +# Gardella-Hirshberg +@article{GardellaHirshberg18, + title={Strongly outer actions of amenable groups on $\mathcal{Z}$-stable {C}*-algebras}, + author={Gardella, Eusebio and Hirshberg, Ilan}, + journal={arXiv preprint arXiv:1811.00447}, + year={2018} +} + +# Giordano-Sierakowski +@article {GiordanoSierakowski16, + AUTHOR = {Giordano, Thierry and Sierakowski, Adam}, + TITLE = {The general linear group as a complete invariant for + {C}*-algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {76}, + YEAR = {2016}, + NUMBER = {2}, + PAGES = {249--269}, + ISSN = {0379-4024}, + MRCLASS = {46L35 (47L80)}, + MRNUMBER = {3552377}, +MRREVIEWER = {Marius V. Ionescu}, + DOI = {10.7900/jot.2015may27.2112}, + URL = {https://doi.org/10.7900/jot.2015may27.2112}, +} + +# Gong +@article {Gong97, + AUTHOR = {Gong, Guihua}, + TITLE = {On inductive limits of matrix algebras over higher-dimensional + spaces. {I}, {II}}, + JOURNAL = {Math. Scand.}, + FJOURNAL = {Mathematica Scandinavica}, + VOLUME = {80}, + YEAR = {1997}, + NUMBER = {1}, + PAGES = {41--55, 56--100}, + ISSN = {0025-5521}, + MRCLASS = {46L05 (19K14 46L35 46L80)}, + MRNUMBER = {1466905}, +MRREVIEWER = {Terry A. Loring}, + DOI = {10.7146/math.scand.a-12611}, + URL = {https://doi.org/10.7146/math.scand.a-12611}, +} + +# Gong-Jiang-Su +@article {GongJiangSu00, + AUTHOR = {Gong, Guihua and Jiang, Xinhui and Su, Hongbing}, + TITLE = {Obstructions to $\mathcal{Z}$-stability for unital simple + {$C^*$}-algebras}, + JOURNAL = {Canad. Math. Bull.}, + FJOURNAL = {Canadian Mathematical Bulletin. Bulletin Canadien de + Math\'{e}matiques}, + VOLUME = {43}, + YEAR = {2000}, + NUMBER = {4}, + PAGES = {418--426}, + ISSN = {0008-4395}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {1793944}, +MRREVIEWER = {Takahiro Sudo}, + DOI = {10.4153/CMB-2000-050-1}, + URL = {https://doi.org/10.4153/CMB-2000-050-1}, +} + +# Gong-Lin-Niu +@article {GongLinNiu20I, + AUTHOR = {Gong, Guihua and Lin, Huaxin and Niu, Zhuang}, + TITLE = {A classification of finite simple amenable $\mathcal{Z}$-stable + {C}*-algebras, {I}: {C}*-algebras with generalized + tracial rank one}, + JOURNAL = {C. R. Math. Acad. Sci. Soc. R. Can.}, + FJOURNAL = {Comptes Rendus Math\'{e}matiques de l'Acad\'{e}mie des Sciences. La + Soci\'{e}t\'{e} Royale du Canada. Mathematical Reports of the Academy + of Science. The Royal Society of Canada}, + VOLUME = {42}, + YEAR = {2020}, + NUMBER = {3}, + PAGES = {63--450}, + ISSN = {0706-1994}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {4215379}, +} + +@article {GongLinNiu20II, + AUTHOR = {Gong, Guihua and Lin, Huaxin and Niu, Zhuang}, + TITLE = {A classification of finite simple amenable $\mathcal{Z}$-stable {C}*-algebras, {II}: {C}*-algebras with rational generalized tracial rank one}, + JOURNAL = {C. R. Math. Acad. Sci. Soc. R. Can.}, + FJOURNAL = {Comptes Rendus Math\'{e}matiques de l'Acad\'{e}mie des Sciences. La + Soci\'{e}t\'{e} Royale du Canada. Mathematical Reports of the Academy + of Science. The Royal Society of Canada}, + VOLUME = {42}, + YEAR = {2020}, + NUMBER = {4}, + PAGES = {451--539}, + ISSN = {0706-1994}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {4215380}, +} + +# Gong-Lin-Xue +@article {GongLinXue15, + AUTHOR = {Gong, Guihua and Lin, Huaxin and Xue, Yifeng}, + TITLE = {Determinant rank of {$C^*$}-algebras}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {274}, + YEAR = {2015}, + NUMBER = {2}, + PAGES = {405--436}, + ISSN = {0030-8730}, + MRCLASS = {46L06 (46L35 46L80)}, + MRNUMBER = {3332910}, +MRREVIEWER = {Claire Anantharaman-Delaroche}, + DOI = {10.2140/pjm.2015.274.405}, + URL = {https://doi.org/10.2140/pjm.2015.274.405}, +} + +# Goodearl +@book {Goodearlbook, + AUTHOR = {Goodearl, Kenneth R.}, + TITLE = {Partially ordered abelian groups with interpolation}, + SERIES = {Mathematical Surveys and Monographs}, + VOLUME = {20}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {1986}, + PAGES = {xxii+336}, + ISBN = {0-8218-1520-2}, + MRCLASS = {06F20 (16A54 18F25 19K14 46A55 46L80)}, + MRNUMBER = {845783}, +MRREVIEWER = {G. A. Elliott}, + DOI = {10.1090/surv/020}, + URL = {https://doi.org/10.1090/surv/020}, +} + +### HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + +# Haagerup +@article {Haagerup87, + AUTHOR = {Haagerup, Uffe}, + TITLE = {{C}onnes' bicentralizer problem and uniqueness of the injective + factor of type {III}$_1$}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {158}, + YEAR = {1987}, + NUMBER = {1-2}, + PAGES = {95--148}, + ISSN = {0001-5962}, + MRCLASS = {46L35}, + MRNUMBER = {880070}, +MRREVIEWER = {Steve Wright}, + DOI = {10.1007/BF02392257}, + URL = {https://doi.org/10.1007/BF02392257}, +} + +# Hatori-Molnar +@article {HatoriMolnar14, + AUTHOR = {Hatori, Osamu and Moln\'{a}r, Lajos}, + TITLE = {Isometries of the unitary groups and {T}hompson isometries of + the spaces of invertible positive elements in + {C}*-algebras}, + JOURNAL = {J. Math. Anal. Appl.}, + FJOURNAL = {Journal of Mathematical Analysis and Applications}, + VOLUME = {409}, + YEAR = {2014}, + NUMBER = {1}, + PAGES = {158--167}, + ISSN = {0022-247X}, + MRCLASS = {46L05}, + MRNUMBER = {3095026}, +MRREVIEWER = {Yangping Jing}, + DOI = {10.1016/j.jmaa.2013.06.065}, + URL = {https://doi.org/10.1016/j.jmaa.2013.06.065}, +} + +# Herman-Ocneanu +@article {HermanOcneanu84, + AUTHOR = {Herman, Richard H. and Ocneanu, Adrian}, + TITLE = {Stability for integer actions on {UHF} {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {59}, + YEAR = {1984}, + NUMBER = {1}, + PAGES = {132--144}, + ISSN = {0022-1236}, + MRCLASS = {46L40}, + MRNUMBER = {763780}, +MRREVIEWER = {Michel Hilsum}, + DOI = {10.1016/0022-1236(84)90056-9}, + URL = {https://doi.org/10.1016/0022-1236(84)90056-9}, +} + +# Higson +@article {Higson88, + AUTHOR = {Higson, Nigel}, + TITLE = {Algebraic {K}-theory of stable {C}*-algebras}, + JOURNAL = {Adv. in Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {67}, + YEAR = {1988}, + NUMBER = {1}, + PAGES = {140}, + ISSN = {0001-8708}, + MRCLASS = {46L80 (19K14 46M20)}, + MRNUMBER = {922140}, +MRREVIEWER = {Cornel Pasnicu}, + DOI = {10.1016/0001-8708(88)90034-5}, + URL = {https://doi.org/10.1016/0001-8708(88)90034-5}, +} + +# Hirshberg-Rordam-Winter +@article {HirshbergRordamWinter07, + AUTHOR = {Hirshberg, Ilan and R{\o}rdam, Mikael and Winter, Wilhelm}, + TITLE = {$\mathcal{C}_0({X})$-algebras, stability and strongly self-absorbing {C}*-algebras}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {339}, + YEAR = {2007}, + NUMBER = {3}, + PAGES = {695--732}, + ISSN = {0025-5831}, + MRCLASS = {46L05 (47L40)}, + MRNUMBER = {2336064}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.1007/s00208-007-0129-8}, + URL = {https://doi.org/10.1007/s00208-007-0129-8}, +} + +# Hirshberg-Orovitz +@article {HirshbergOrovitz13, + AUTHOR = {Hirshberg, Ilan and Orovitz, Joav}, + TITLE = {Tracially $\mathcal{Z}$-absorbing {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {265}, + YEAR = {2013}, + NUMBER = {5}, + PAGES = {765--785}, + ISSN = {0022-1236}, + MRCLASS = {46L05}, + MRNUMBER = {3063095}, +MRREVIEWER = {Stuart A. White}, + DOI = {10.1016/j.jfa.2013.05.005}, + URL = {https://doi.org/10.1016/j.jfa.2013.05.005}, +} + +# Hirshberg-Winter +@article {HirshbergWinter07, + AUTHOR = {Hirshberg, Ilan and Winter, Wilhelm}, + TITLE = {{R}okhlin actions and self-absorbing {C}*-algebras}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {233}, + YEAR = {2007}, + NUMBER = {1}, + PAGES = {125--143}, + ISSN = {0030-8730}, + MRCLASS = {46L05 (46L55)}, + MRNUMBER = {2366371}, +MRREVIEWER = {Efren Ruiz}, + DOI = {10.2140/pjm.2007.233.125}, + URL = {https://doi.org/10.2140/pjm.2007.233.125}, +} + +@article {HirshbergWinter08, + AUTHOR = {Hirshberg, Ilan and Winter, Wilhelm}, + TITLE = {Permutations of strongly self-absorbing {C}*-algebras}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {19}, + YEAR = {2008}, + NUMBER = {9}, + PAGES = {1137--1145}, + ISSN = {0129-167X}, + MRCLASS = {46L05 (46L45 46L55)}, + MRNUMBER = {2458564}, +MRREVIEWER = {Yoshikazu Katayama}, + DOI = {10.1142/S0129167X08005011}, + URL = {https://doi.org/10.1142/S0129167X08005011}, +} + +# Husemoller +@book {Husemoller66, + AUTHOR = {Husemoller, Dale}, + TITLE = {Fibre bundles}, + PUBLISHER = {McGraw-Hill Book Co., New York-London-Sydney}, + YEAR = {1966}, + PAGES = {xiv+300}, + MRCLASS = {57.30 (55.00)}, + MRNUMBER = {0229247}, +MRREVIEWER = {R. Williamson}, +} + +# Hurewicz-Wallman +@book {HurewiczWallman, + AUTHOR = {Hurewicz, Witold and Wallman, Henry}, + TITLE = {Dimension theory}, + SERIES = {Princeton Mathematical Series, vol. 4}, + PUBLISHER = {Princeton University Press, Princeton, N. J.}, + YEAR = {1941}, + PAGES = {vii+165}, + MRCLASS = {56.0X}, + MRNUMBER = {0006493}, +MRREVIEWER = {H. Whitney}, +} + + +### IIIIIIIIIIIIIIIIIIIIIIIIIIIIII + +# Izumi +@article{Izumi02, + title={Inclusions of simple {C}*-algebras}, + author={Izumi, Masaki}, + year={2002}, + volume = {547}, + pages = {97--138}, + journal = {J. Reine Angew. Math.}, + publisher={Walter de Gruyter GmbH \& Co. KG Berlin, Germany}, +} + +@article {Izumi04, + AUTHOR = {Izumi, Masaki}, + TITLE = {Finite group actions on {C}*-algebras with the {R}ohlin property. {I}}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {122}, + YEAR = {2004}, + NUMBER = {2}, + PAGES = {233--280}, + ISSN = {0012-7094}, + MRCLASS = {46L55 (19K35 46L35 46L40 46L80)}, + MRNUMBER = {2053753}, +MRREVIEWER = {Valentin Deaconu}, + DOI = {10.1215/S0012-7094-04-12221-3}, + URL = {https://doi.org/10.1215/S0012-7094-04-12221-3}, +} + +### JJJJJJJJJJJJJJJJJJJJJ +# Jiang +@article{Jiang97, + title={Nonstable {K}-theory for $\mathcal{Z}$-stable {C}*-algebras}, + author={Jiang, Xinhui}, + journal={arXiv preprint math/9707228}, + year={1997} +} + +# Jiang-Su +@article {JiangSu99, + AUTHOR = {Jiang, Xinhui and Su, Hongbing}, + TITLE = {On a simple unital projectionless {C}*-algebra}, + JOURNAL = {Amer. J. Math.}, + FJOURNAL = {American Journal of Mathematics}, + VOLUME = {121}, + YEAR = {1999}, + NUMBER = {2}, + PAGES = {359--413}, + ISSN = {0002-9327}, + MRCLASS = {46L35 (19K35 46L80)}, + MRNUMBER = {1680321}, +MRREVIEWER = {Vicumpriya S. Perera}, + URL = + {http://muse.jhu.edu/journals/american_journal_of_mathematics/v121/121.2jiang.pdf}, +} + +#Jolissaint +@article{Jolissaint16, + AUTHOR = {Jolissaint, Paul}, + TITLE = {Relative inner amenability and relative property gamma}, + JOURNAL = {Math. Scand.}, + FJOURNAL = {Mathematica Scandinavica}, + VOLUME = {119}, + YEAR = {2016}, + NUMBER = {2}, + PAGES = {293--319}, + ISSN = {0025-5521}, + MRCLASS = {22D25 (22D10 22F05 43A07 46L37)}, + MRNUMBER = {3570949}, +MRREVIEWER = {Jonathan M. Rosenberg}, + DOI = {10.7146/math.scand.a-24748}, + URL = {https://doi.org/10.7146/math.scand.a-24748}, +} + +### KKKKKKKKKKKKKKKKKKKKKKKKK +# Kawamuro +@article {Kawamuro99, + AUTHOR = {Kawamuro, Keiko}, + TITLE = {Central sequence subfactors and double commutant properties}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {10}, + YEAR = {1999}, + NUMBER = {1}, + PAGES = {53--77}, + ISSN = {0129-167X}, + MRCLASS = {46L37}, + MRNUMBER = {1678538}, +MRREVIEWER = {Carl Winsl{\o}w}, + DOI = {10.1142/S0129167X99000033}, + URL = {https://doi.org/10.1142/S0129167X99000033}, +} + +# Kirchberg +@incollection {Kirchberg06, + AUTHOR = {Kirchberg, Eberhard}, + TITLE = {Central sequences in {C}*-algebras and strongly purely + infinite algebras}, + BOOKTITLE = {Operator {A}lgebras: {T}he {A}bel {S}ymposium 2004}, + SERIES = {Abel Symp.}, + VOLUME = {1}, + PAGES = {175--231}, + PUBLISHER = {Springer, Berlin}, + YEAR = {2006}, + MRCLASS = {46L05}, + MRNUMBER = {2265050}, +MRREVIEWER = {Wilhelm Winter}, + DOI = {10.1007/978-3-540-34197-0\_10}, + URL = {https://doi.org/10.1007/978-3-540-34197-0_10}, +} + +# Kirchberg-Phillips +@article {KirchbergPhillips00, + AUTHOR = {Kirchberg, Eberhard and Phillips, N. Christopher}, + TITLE = {Embedding of exact {C}*-algebras in the {C}untz algebra $\mathcal{O}_2$}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {525}, + YEAR = {2000}, + PAGES = {17--53}, + ISSN = {0075-4102}, + MRCLASS = {46L05 (19K56 46L35 46L80)}, + MRNUMBER = {1780426}, +MRREVIEWER = {Mikael R{\o}rdam}, + DOI = {10.1515/crll.2000.065}, + URL = {https://doi.org/10.1515/crll.2000.065}, +} + +# Kishimoto +@article {Kishimoto00, + AUTHOR = {Kishimoto, A.}, + TITLE = {{R}ohlin property for shift automorphisms}, + JOURNAL = {Rev. Math. Phys.}, + FJOURNAL = {Reviews in Mathematical Physics. A Journal for Both Review and + Original Research Papers in the Field of Mathematical Physics}, + VOLUME = {12}, + YEAR = {2000}, + NUMBER = {7}, + PAGES = {965--980}, + ISSN = {0129-055X}, + MRCLASS = {46L40 (46L05 46L55)}, + MRNUMBER = {1782691}, +MRREVIEWER = {Claire Anantharaman-Delaroche}, + DOI = {10.1142/S0129055X00000368}, + URL = {https://doi.org/10.1142/S0129055X00000368}, +} + +# Kumjian +@article {Kumjian88, + AUTHOR = {Kumjian, Alexander}, + TITLE = {An involutive automorphism of the {B}unce-{D}eddens algebra}, + JOURNAL = {C. R. Math. Rep. Acad. Sci. Canada}, + FJOURNAL = {La Soci\'{e}t\'{e} Royale du Canada. L'Academie des Sciences. Comptes + Rendus Math\'{e}matiques. (Mathematical Reports)}, + VOLUME = {10}, + YEAR = {1988}, + NUMBER = {5}, + PAGES = {217--218}, + ISSN = {0706-1994}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {962104}, +MRREVIEWER = {J. W. Bunce}, +} + +### LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL +# Luck-Rordam +@article {LuckRordam93, + AUTHOR = {L\"{u}ck, Wolfgang and R{\o}rdam, Mikael}, + TITLE = {Algebraic {K}-theory of von {N}eumann algebras}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {7}, + YEAR = {1993}, + NUMBER = {6}, + PAGES = {517--536}, + ISSN = {0920-3036}, + MRCLASS = {46L80 (19B28 19J10 19K56 19K99)}, + MRNUMBER = {1268591}, +MRREVIEWER = {Jonathan M. Rosenberg}, + DOI = {10.1007/BF00961216}, + URL = {https://doi.org/10.1007/BF00961216}, +} + +### MMMMMMMMMMMMMMMMMMMMMMM +# Maclane +@book {Maclane12, + AUTHOR = {MacLane, Saunders}, + TITLE = {Homology}, + SERIES = {Die Grundlehren der mathematischen Wissenschaften, Band 114}, + EDITION = {first}, + PUBLISHER = {Springer-Verlag, Berlin-New York}, + YEAR = {1967}, + PAGES = {x+422}, + MRCLASS = {18-02}, + MRNUMBER = {0349792}, +} + +# Matui-Sato +@article {MatuiSato12, + AUTHOR = {Matui, Hiroki and Sato, Yasuhiko}, + TITLE = {Strict comparison and $\mathcal{Z}$-absorption of nuclear {C}*-algebras}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {209}, + YEAR = {2012}, + NUMBER = {1}, + PAGES = {179--196}, + ISSN = {0001-5962}, + MRCLASS = {46L05}, + MRNUMBER = {2979512}, +MRREVIEWER = {Caleb Eckhardt}, + DOI = {10.1007/s11511-012-0084-4}, + URL = {https://doi.org/10.1007/s11511-012-0084-4}, +} + +#McDuff +@article {McDuff69, + AUTHOR = {McDuff, Dusa}, + TITLE = {Uncountably many {II}$_1$ factors}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {90}, + YEAR = {1969}, + PAGES = {372--377}, + ISSN = {0003-486X}, + MRCLASS = {46.65}, + MRNUMBER = {259625}, +MRREVIEWER = {Z. Takeda}, + DOI = {10.2307/1970730}, + URL = {https://doi.org/10.2307/1970730}, +} + +@article {McDuff70, + AUTHOR = {McDuff, Dusa}, + TITLE = {Central sequences and the hyperfinite factor}, + JOURNAL = {Proc. London Math. Soc. (3)}, + FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, + VOLUME = {21}, + YEAR = {1970}, + PAGES = {443--461}, + ISSN = {0024-6115}, + MRCLASS = {46.65}, + MRNUMBER = {281018}, +MRREVIEWER = {M. Takesaki}, + DOI = {10.1112/plms/s3-21.3.443}, + URL = {https://doi.org/10.1112/plms/s3-21.3.443}, +} + +# Munkres +@book {Munkres, + AUTHOR = {Munkres, James R.}, + TITLE = {Topology}, + NOTE = {Second edition of [ MR0464128]}, + PUBLISHER = {Prentice Hall, Inc., Upper Saddle River, NJ}, + YEAR = {2000}, + PAGES = {xvi+537}, + ISBN = {0-13-181629-2}, + MRCLASS = {54-01}, + MRNUMBER = {3728284}, +} + +# Murphy +@book {Murphybook, + AUTHOR = {Murphy, Gerard J.}, + TITLE = {{C}*-algebras and operator theory}, + PUBLISHER = {Academic Press, Inc., Boston, MA}, + YEAR = {1990}, + PAGES = {x+286}, + ISBN = {0-12-511360-9}, + MRCLASS = {46Lxx (46-01)}, + MRNUMBER = {1074574}, +MRREVIEWER = {E. Gerlach}, +} + +# Murray-von Neumann +@article {MvNIV, + AUTHOR = {Murray, Francis J. and von Neumann, John}, + TITLE = {On rings of operators. {IV}}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {44}, + YEAR = {1943}, + PAGES = {716--808}, + ISSN = {0003-486X}, + MRCLASS = {46.0X}, + MRNUMBER = {9096}, +MRREVIEWER = {E. R. Lorch}, + DOI = {10.2307/1969107}, + URL = {https://doi.org/10.2307/1969107}, +} + +# Mygind +@article{Mygind01, + title={Classification of certain simple {C}*-algebras with torsion in {K}1}, + author={Mygind, Jesper}, + journal={Canadian Journal of Mathematics}, + volume={53}, + number={6}, + pages={1223--1308}, + year={2001}, + publisher={Cambridge University Press} +} +@article {MR1863849, + AUTHOR = {Mygind, Jesper}, + TITLE = {Classification of certain simple {C}*-algebras with torsion in {$K_1$}}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {53}, + YEAR = {2001}, + NUMBER = {6}, + PAGES = {1223--1308}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (19K14 46L80)}, + MRNUMBER = {1863849}, +MRREVIEWER = {Xiao Chun Fang}, + DOI = {10.4153/CJM-2001-046-2}, + URL = {https://doi.org/10.4153/CJM-2001-046-2}, +} + +### NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +# Ng +@article {Ng14, + AUTHOR = {Ng, Ping W.}, + TITLE = {The kernel of the determinant map on certain simple {C}*-algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {71}, + YEAR = {2014}, + NUMBER = {2}, + PAGES = {341--379}, + ISSN = {0379-4024}, + MRCLASS = {46L05 (46L80 47B47 47C15)}, + MRNUMBER = {3214642}, +MRREVIEWER = {Takahiro Sudo}, + DOI = {10.7900/jot.2012apr01.1953}, + URL = {https://doi.org/10.7900/jot.2012apr01.1953}, +} + +# Ng-Robert +@article{NgRobert15, + title={Sums of commutators in pure {C}*-algebras}, + author={Ng, Ping W. and Robert, Leonel}, + journal={arXiv preprint arXiv:1504.00046}, + year={2015} +} + +@article {NgRobert17, + AUTHOR = {Ng, Ping W. and Robert, Leonel}, + TITLE = {The kernel of the determinant map on pure {C}*-algebras}, + JOURNAL = {Houston J. Math.}, + FJOURNAL = {Houston Journal of Mathematics}, + VOLUME = {43}, + YEAR = {2017}, + NUMBER = {1}, + PAGES = {139--168}, + ISSN = {0362-1588}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {3647937}, +MRREVIEWER = {Daniele Puglisi}, + DOI = {10.1007/s11139-016-9879-9}, + URL = {https://doi.org/10.1007/s11139-016-9879-9}, +} + +# Nielsen-Thomsen +@article {NielsenThomsen96, + AUTHOR = {Nielsen, Karen E. and Thomsen, Klaus}, + TITLE = {Limits of circle algebras}, + JOURNAL = {Exposition. Math.}, + FJOURNAL = {Expositiones Mathematicae. International Journal}, + VOLUME = {14}, + YEAR = {1996}, + NUMBER = {1}, + PAGES = {17--56}, + ISSN = {0723-0869}, + MRCLASS = {46L80 (19K14 46L05)}, + MRNUMBER = {1382013}, +MRREVIEWER = {Sze-Kai Tsui}, +} + +### OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO + +@article {OsakaTeruya18, + AUTHOR = {Osaka, Hiroyuki and Teruya, Tamotsu}, + TITLE = {The {J}iang--{S}u absorption for inclusions of unital {C}*-algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {70}, + YEAR = {2018}, + NUMBER = {2}, + PAGES = {400--425}, + ISSN = {0008-414X}, + MRCLASS = {46L55 (46L35)}, + MRNUMBER = {3759005}, +MRREVIEWER = {Daniele Puglisi}, + DOI = {10.4153/CJM-2017-033-7}, + URL = {https://doi.org/10.4153/CJM-2017-033-7}, +} + +### PPPPPPPPPPPPPPPPPPPPPPP +# Paterson +@article{Paterson83, + AUTHOR = {Paterson, Alan L. T.}, + TITLE = {Harmonic analysis on unitary groups}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {53}, + YEAR = {1983}, + NUMBER = {3}, + PAGES = {203--223}, + ISSN = {0022-1236}, + MRCLASS = {22D10 (22D25 43A65 46L05)}, + MRNUMBER = {724026}, +MRREVIEWER = {J. Gil de Lamadrid}, + DOI = {10.1016/0022-1236(83)90031-9}, + URL = {https://doi.org/10.1016/0022-1236(83)90031-9}, +} + +# Paulsen +@book{Paulsenbook, + AUTHOR = {Paulsen, Vern}, + TITLE = {Completely bounded maps and operator algebras}, + SERIES = {Cambridge Studies in Advanced Mathematics}, + VOLUME = {78}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {2002}, + PAGES = {xii+300}, + ISBN = {0-521-81669-6}, + MRCLASS = {46L07 (47A20 47L30)}, + MRNUMBER = {1976867}, +MRREVIEWER = {Christian Le Merdy}, +} + +# Phillips +@article {Phillips92, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {The rectifiable metric on the space of projections in a {C}*-algebra}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {3}, + YEAR = {1992}, + NUMBER = {5}, + PAGES = {679--698}, + ISSN = {0129-167X}, + MRCLASS = {46L05}, + MRNUMBER = {1189681}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1142/S0129167X92000333}, + URL = {https://doi.org/10.1142/S0129167X92000333}, +} + +@article {Phillips95, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {Exponential length and traces}, + JOURNAL = {Proc. Roy. Soc. Edinburgh Sect. A}, + FJOURNAL = {Proceedings of the Royal Society of Edinburgh. Section A. + Mathematics}, + VOLUME = {125}, + YEAR = {1995}, + NUMBER = {1}, + PAGES = {13--29}, + ISSN = {0308-2105}, + MRCLASS = {46L05}, + MRNUMBER = {1318621}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1017/S0308210500030730}, + URL = {https://doi.org/10.1017/S0308210500030730}, +} + +@article {Phillips00, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {A classification theorem for nuclear purely infinite simple {C}*-algebras}, + JOURNAL = {Doc. Math.}, + FJOURNAL = {Documenta Mathematica}, + VOLUME = {5}, + YEAR = {2000}, + PAGES = {49--114}, + ISSN = {1431-0635}, + MRCLASS = {46L05 (19K56 46L35 46L80)}, + MRNUMBER = {1745197}, +MRREVIEWER = {Mikael R{\o}rdam}, +} + +@article {Phillips01, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {Continuous-trace {C}*-algebras not isomorphic to their opposite algebras}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {12}, + YEAR = {2001}, + NUMBER = {3}, + PAGES = {263--275}, + ISSN = {0129-167X}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {1841515}, +MRREVIEWER = {Takahiro Sudo}, + DOI = {10.1142/S0129167X01000642}, + URL = {https://doi.org/10.1142/S0129167X01000642}, +} + +@article{Phillips04, + title={A simple separable {C}*-algebra not isomorphic to its opposite algebra}, + author={Phillips, N. Christopher}, + journal={Proceedings of the American Mathematical Society}, + volume={132}, + number={10}, + pages={2997--3005}, + year={2004} +} +@article {Phillips04, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {A simple separable {C}*-algebra not isomorphic to its opposite algebra}, + JOURNAL = {Proc. Amer. Math. Soc.}, + FJOURNAL = {Proceedings of the American Mathematical Society}, + VOLUME = {132}, + YEAR = {2004}, + NUMBER = {10}, + PAGES = {2997--3005}, + ISSN = {0002-9939}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {2063121}, +MRREVIEWER = {Hiroyuki Osaka}, + DOI = {10.1090/S0002-9939-04-07330-7}, + URL = {https://doi.org/10.1090/S0002-9939-04-07330-7}, +} + +@article{Phillips12, + title={The tracial {R}okhlin property is generic}, + author={Phillips, N. Christopher}, + journal={arXiv preprint arXiv:1209.3859}, + year={2012} +} + +# Phillips-Viola +@article {PhillipsViola13, + AUTHOR = {Phillips, N. Christopher and Viola, Maria Grazia}, + TITLE = {A simple separable exact {C}*-algebra not anti-isomorphic to itself}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {355}, + YEAR = {2013}, + NUMBER = {2}, + PAGES = {783--799}, + ISSN = {0025-5831}, + MRCLASS = {46L35 (46L09 46L37 46L40)}, + MRNUMBER = {3010147}, +MRREVIEWER = {Snigdhayan Mahanta}, + DOI = {10.1007/s00208-011-0755-z}, + URL = {https://doi.org/10.1007/s00208-011-0755-z}, +} + +# Pimsner-Voiculescu +@article {PimsnerVoiculescu80, + AUTHOR = {Pimsner, Mihai and Voiculescu, Dan}, + TITLE = {Imbedding the irrational rotation {C}*-algebra into {AF}-algebra}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {4}, + YEAR = {1980}, + NUMBER = {2}, + PAGES = {201--210}, + ISSN = {0379-4024}, + MRCLASS = {46L35 (16A54)}, + MRNUMBER = {595412}, +MRREVIEWER = {David Handelman}, +} + +# Popa +@article {Popa83, + AUTHOR = {Popa, Sorin}, + TITLE = {Orthogonal pairs of *-subalgebras in finite von {N}eumann algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {9}, + YEAR = {1983}, + NUMBER = {2}, + PAGES = {253--268}, + ISSN = {0379-4024}, + MRCLASS = {46L10 (16A30 20C07)}, + MRNUMBER = {703810}, +MRREVIEWER = {G. A. Elliott}, +} + + +@article {Popa89, + AUTHOR = {Popa, Sorin}, + TITLE = {Sousfacteurs, actions des groupes et cohomologie}, + JOURNAL = {C. R. Acad. Sci. Paris S\'{e}r. I Math.}, + FJOURNAL = {Comptes Rendus de l'Acad\'{e}mie des Sciences. S\'{e}rie I. + Math\'{e}matique}, + VOLUME = {309}, + YEAR = {1989}, + NUMBER = {12}, + PAGES = {771--776}, + ISSN = {0764-4442}, + MRCLASS = {46L37 (22D25 22D35 46L55 46L80)}, + MRNUMBER = {1054961}, +MRREVIEWER = {Masatoshi Enomoto}, +} + +@article {Popa00, + AUTHOR = {Popa, Sorin}, + TITLE = {On the relative {D}ixmier property for inclusions of {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {171}, + YEAR = {2000}, + NUMBER = {1}, + PAGES = {139--154}, + ISSN = {0022-1236}, + MRCLASS = {46L37 (46L05)}, + MRNUMBER = {1742862}, +MRREVIEWER = {H. Halpern}, + DOI = {10.1006/jfan.1999.3536}, + URL = {https://doi.org/10.1006/jfan.1999.3536}, +} + +### RRRRRRRRRRRRRRRRRRRRRRRRRR +# Raeburn +@book {RaeburnBook, + AUTHOR = {Raeburn, Iain}, + TITLE = {Graph algebras}, + SERIES = {CBMS Regional Conference Series in Mathematics}, + VOLUME = {103}, + PUBLISHER = {Published for the Conference Board of the Mathematical + Sciences, Washington, DC; by the American Mathematical + Society, Providence, RI}, + YEAR = {2005}, + PAGES = {vi+113}, + ISBN = {0-8218-3660-9}, + MRCLASS = {46L05 (22D25)}, + MRNUMBER = {2135030}, +MRREVIEWER = {Mark Tomforde}, + DOI = {10.1090/cbms/103}, + URL = {https://doi.org/10.1090/cbms/103}, +} + +#Rieffel +@article {Rieffel87, + AUTHOR = {Rieffel, Marc A.}, + TITLE = {The homotopy groups of the unitary groups of noncommutative tori}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {17}, + YEAR = {1987}, + NUMBER = {2}, + PAGES = {237--254}, + ISSN = {0379-4024}, + MRCLASS = {22D25 (46L55 46L80)}, + MRNUMBER = {887221}, +MRREVIEWER = {Gustavo Corach}, +} + +# Rordam +@article {Rordam91, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {On the structure of simple {C}*-algebras tensored with a {UHF}-algebra}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {100}, + YEAR = {1991}, + NUMBER = {1}, + PAGES = {1--17}, + ISSN = {0022-1236}, + MRCLASS = {46L05}, + MRNUMBER = {1124289}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.1016/0022-1236(91)90098-P}, + URL = {https://doi.org/10.1016/0022-1236(91)90098-P}, +} + + +@article {Rordam92, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {On the structure of simple {C}*-algebras tensored with a {UHF}-algebra. {II}}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {107}, + YEAR = {1992}, + NUMBER = {2}, + PAGES = {255--269}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L85)}, + MRNUMBER = {1172023}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.1016/0022-1236(92)90106-S}, + URL = {https://doi.org/10.1016/0022-1236(92)90106-S}, +} + +@article {Rordam93, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Classification of inductive limits of {C}untz algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {440}, + YEAR = {1993}, + PAGES = {175--200}, + ISSN = {0075-4102}, + MRCLASS = {46L05 (19K99 46L80)}, + MRNUMBER = {1225963}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1515/crll.1993.440.175}, + URL = {https://doi.org/10.1515/crll.1993.440.175}, +} + +@article {Rordam95, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Classification of certain infinite simple {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {131}, + YEAR = {1995}, + NUMBER = {2}, + PAGES = {415--458}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (19K35 46L35 46L80)}, + MRNUMBER = {1345038}, +MRREVIEWER = {Judith A. Packer}, + DOI = {10.1006/jfan.1995.1095}, + URL = {https://doi.org/10.1006/jfan.1995.1095}, +} + +@incollection {RordamBook, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Classification of nuclear, simple {C}*-algebras}, + BOOKTITLE = {Classification of nuclear {C}*-algebras. {E}ntropy in + operator algebras}, + SERIES = {Encyclopaedia Math. Sci.}, + VOLUME = {126}, + PAGES = {1--145}, + PUBLISHER = {Springer, Berlin}, + YEAR = {2002}, + MRCLASS = {46L05 (19K56 46L35 46L80)}, + MRNUMBER = {1878882}, +MRREVIEWER = {Judith A. Packer}, + DOI = {10.1007/978-3-662-04825-2\_1}, + URL = {https://doi.org/10.1007/978-3-662-04825-2_1}, +} + +@book {RordamKBook, + AUTHOR = {R{\o}rdam, Mikael and Larsen, Flemming and Laustsen, Niels J.}, + TITLE = {An introduction to {K}-theory for {C}*-algebras}, + SERIES = {London Mathematical Society Student Texts}, + VOLUME = {49}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {2000}, + PAGES = {xii+242}, + ISBN = {0-521-78334-8; 0-521-78944-3}, + MRCLASS = {46-01 (19K35 46L80)}, + MRNUMBER = {1783408}, +MRREVIEWER = {\'{E}ric Leichtnam}, +} + +@article {Rordam03, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {A simple {C}*-algebra with a finite and an infinite projection}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {191}, + YEAR = {2003}, + NUMBER = {1}, + PAGES = {109--142}, + ISSN = {0001-5962}, + MRCLASS = {46L05}, + MRNUMBER = {2020420}, +MRREVIEWER = {Berndt Brenken}, + DOI = {10.1007/BF02392697}, + URL = {https://doi.org/10.1007/BF02392697}, +} + +@article {Rordam04, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {The stable and the real rank of $\mathcal{Z}$-absorbing {C}*-algebras}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {15}, + YEAR = {2004}, + NUMBER = {10}, + PAGES = {1065--1084}, + ISSN = {0129-167X}, + MRCLASS = {46L35 (19K14 46L05 46L06)}, + MRNUMBER = {2106263}, +MRREVIEWER = {Nadia S. Larsen}, + DOI = {10.1142/S0129167X04002661}, + URL = {https://doi.org/10.1142/S0129167X04002661}, +} + +@article{Rordam21, + title={Irreducible inclusions of simple {C}*-algebras}, + author={R{\o}rdam, Mikael}, + journal={arXiv preprint arXiv:2105.11899}, + year={2021} +} + +# Rordam-Winter +@article {RordamWinter10, + AUTHOR = {R{\o}rdam, Mikael and Winter, Wilhelm}, + TITLE = {The {J}iang-{S}u algebra revisited}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {642}, + YEAR = {2010}, + PAGES = {129--155}, + ISSN = {0075-4102}, + MRCLASS = {46L35 (46L05 46L85)}, + MRNUMBER = {2658184}, +MRREVIEWER = {Hiroyuki Osaka}, + DOI = {10.1515/CRELLE.2010.039}, + URL = {https://doi.org/10.1515/CRELLE.2010.039}, +} + +# Rosenberg +@article {Rosenberg89, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {Continuous-trace algebras from the bundle theoretic point of + view}, + JOURNAL = {J. Austral. Math. Soc. Ser. A}, + FJOURNAL = {Australian Mathematical Society. Journal. Series A. Pure + Mathematics and Statistics}, + VOLUME = {47}, + YEAR = {1989}, + NUMBER = {3}, + PAGES = {368--381}, + ISSN = {0263-6115}, + MRCLASS = {46L80 (19K99 46L10 46M20 55R10)}, + MRNUMBER = {1018964}, +MRREVIEWER = {Claude Schochet}, +} + +@incollection {Rosenberg04, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {Comparison between algebraic and topological {K}-theory for + {B}anach algebras and {C}*-algebras}, + BOOKTITLE = {Handbook of {$K$}-theory. {V}ol. 1, 2}, + PAGES = {843--874}, + PUBLISHER = {Springer, Berlin}, + YEAR = {2005}, + MRCLASS = {46L80 (19K99 46H99)}, + MRNUMBER = {2181834}, +MRREVIEWER = {Efton Park}, + DOI = {10.1007/978-3-540-27855-9\_16}, + URL = {https://doi.org/10.1007/978-3-540-27855-9_16}, +} + + +@article {Rosenberg97, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {The algebraic {$K$}-theory of operator algebras}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {12}, + YEAR = {1997}, + NUMBER = {1}, + PAGES = {75--99}, + ISSN = {0920-3036}, + MRCLASS = {19K99 (19D35 19D50 19L41 46L80)}, + MRNUMBER = {1466624}, +MRREVIEWER = {Hiroshi Takai}, + DOI = {10.1023/A:1007736420938}, + URL = {https://doi.org/10.1023/A:1007736420938}, +} + +@book {Rosenberg94, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {Algebraic {$K$}-theory and its applications}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {147}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1994}, + PAGES = {x+392}, + ISBN = {0-387-94248-3}, + MRCLASS = {19-01 (19-02)}, + MRNUMBER = {1282290}, +MRREVIEWER = {Dominique Arlettaz}, + DOI = {10.1007/978-1-4612-4314-4}, + URL = {https://doi.org/10.1007/978-1-4612-4314-4}, +} + +# Rosenberg-Schochet +@article {RosenbergSchochet87, + AUTHOR = {Rosenberg, Jonathan and Schochet, Claude}, + TITLE = {The {K}\"{u}nneth theorem and the universal coefficient theorem + for {K}asparov's generalized {$K$}-functor}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {55}, + YEAR = {1987}, + NUMBER = {2}, + PAGES = {431--474}, + ISSN = {0012-7094}, + MRCLASS = {46L80 (19K33 46M20 58G12)}, + MRNUMBER = {894590}, +MRREVIEWER = {Thierry Fack}, + DOI = {10.1215/S0012-7094-87-05524-4}, + URL = {https://doi.org/10.1215/S0012-7094-87-05524-4}, +} + + +### SSSSSSSSSSSSSSSSSSS +# Sakai +@article {Sakai55, + AUTHOR = {Sakai, Sh\^{o}ichir\^{o}}, + TITLE = {On the group isomorphism of unitary groups in {AW}*-algebras}, + JOURNAL = {Tohoku Math. J. (2)}, + FJOURNAL = {The Tohoku Mathematical Journal. Second Series}, + VOLUME = {7}, + YEAR = {1955}, + PAGES = {87--95}, + ISSN = {0040-8735}, + MRCLASS = {46.2X}, + MRNUMBER = {73139}, +MRREVIEWER = {J. Feldman}, + DOI = {10.2748/tmj/1178245106}, + URL = {https://doi.org/10.2748/tmj/1178245106}, +} + +@article {Sakai70, + AUTHOR = {Sakai, Sh\^{o}ichir\^{o}}, + TITLE = {An uncountable number of {II}$_1$ and {II}$_{\infty}$ factors}, + JOURNAL = {J. Functional Analysis}, + VOLUME = {5}, + YEAR = {1970}, + PAGES = {236--246}, + MRCLASS = {46.65}, + MRNUMBER = {0259626}, +MRREVIEWER = {Z. Takeda}, + DOI = {10.1016/0022-1236(70)90028-5}, + URL = {https://doi.org/10.1016/0022-1236(70)90028-5}, +} + +@book{Sakaibook, + title={{C}*-algebras and {W}*-algebras}, + author={Sakai, Sh\^{o}ichir\^{o}}, + year={2012}, + publisher={Springer Science \& Business Media} +} + +# Sarkowicz +@article{Sarkowicz, + title={Tensorially absorbing inclusions of {C}*-algebras}, + author={Sarkowicz, Pawel}, + journal={arXiv preprint arXiv:2211.14974}, + year={2022}, + NOTE={preprint} +} + +# Sarkowicz-Tikuisis +@misc{SarkowiczTikuisis, + title={Polar decomposition in algebraic {K}-theory}, + author={Pawel Sarkowicz and Aaron Tikuisis}, + year={2023}, + journal={arXiv preprint arXiv:2303.16248}, + NOTE={preprint} +} + +# Sato +@article {Sato10, + AUTHOR = {Sato, Yasuhiko}, + TITLE = {The {R}ohlin property for automorphisms of the {J}iang-{S}u + algebra}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {259}, + YEAR = {2010}, + NUMBER = {2}, + PAGES = {453--476}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L35 46L40 46L80)}, + MRNUMBER = {2644109}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.1016/j.jfa.2010.04.006}, + URL = {https://doi.org/10.1016/j.jfa.2010.04.006}, +} + +# Sato-White-Winter +@article {SatoWhiteWinter15, + AUTHOR = {Sato, Yasuhiko and White, Stuart and Winter, Wilhelm}, + TITLE = {Nuclear dimension and {$\mathcal{Z}$}-stability}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {202}, + YEAR = {2015}, + NUMBER = {2}, + PAGES = {893--921}, + ISSN = {0020-9910}, + MRCLASS = {46L35 (45L05)}, + MRNUMBER = {3418247}, +MRREVIEWER = {Aaron Tikuisis}, + DOI = {10.1007/s00222-015-0580-1}, + URL = {https://doi.org/10.1007/s00222-015-0580-1}, +} + +# Schafhauser +@article {Schafhauser20, + AUTHOR = {Schafhauser, Christopher}, + TITLE = {Subalgebras of simple {AF}-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {192}, + YEAR = {2020}, + NUMBER = {2}, + PAGES = {309--352}, + ISSN = {0003-486X}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {4151079}, +MRREVIEWER = {Daniele Puglisi}, + DOI = {10.4007/annals.2020.192.2.1}, + URL = {https://doi.org/10.4007/annals.2020.192.2.1}, +} + +# Schemaitat +@article {Schemaitat22, + AUTHOR = {Schemaitat, Andr\'{e}}, + TITLE = {The {J}iang-{S}u algebra is strongly self-absorbing revisited}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {282}, + YEAR = {2022}, + NUMBER = {6}, + PAGES = {Paper No. 109347, 39}, + ISSN = {0022-1236}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {4360358}, +MRREVIEWER = {Prahlad Vaidyanathan}, + DOI = {10.1016/j.jfa.2021.109347}, + URL = {https://doi.org/10.1016/j.jfa.2021.109347}, +} + +# Schochet +@article {SchochetIV, + AUTHOR = {Schochet, Claude}, + TITLE = {Topological methods for {C}*-algebras. {IV}. {M}od {$p$} homology}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {114}, + YEAR = {1984}, + NUMBER = {2}, + PAGES = {447--468}, + ISSN = {0030-8730}, + MRCLASS = {46L80 (19K33 46M20 55N99)}, + MRNUMBER = {757511}, +MRREVIEWER = {Vern Paulsen}, + URL = {http://projecteuclid.org/euclid.pjm/1102708718}, +} + +# Serre +@book {Serre77, + AUTHOR = {Serre, Jean-Pierre}, + TITLE = {Linear representations of finite groups}, + SERIES = {Graduate Texts in Mathematics, Vol. 42}, + NOTE = {Translated from the second French edition by Leonard L. Scott}, + PUBLISHER = {Springer-Verlag, New York-Heidelberg}, + YEAR = {1977}, + PAGES = {x+170}, + ISBN = {0-387-90190-6}, + MRCLASS = {20CXX}, + MRNUMBER = {0450380}, +MRREVIEWER = {W. Feit}, +} + +# Silverman +@book{Silverman14, + title={A friendly introduction to number theory}, + author={Silverman, Joseph H.}, + year={2014}, + publisher={Pearson} +} + +### TTTTTTTTTTTTTTTTTTTT + +# Tatsuuma-Shimomura-Hirai +@article {TatsuumaShimomuraHirai98, + AUTHOR = {Tatsuuma, Nobuhiko and Shimomura, Hiroaki and Hirai, Takeshi}, + TITLE = {On group topologies and unitary representations of inductive + limits of topological groups and the case of the group of + diffeomorphisms}, + JOURNAL = {J. Math. Kyoto Univ.}, + FJOURNAL = {Journal of Mathematics of Kyoto University}, + VOLUME = {38}, + YEAR = {1998}, + NUMBER = {3}, + PAGES = {551--578}, + ISSN = {0023-608X}, + MRCLASS = {22A05 (54H11 58D05)}, + MRNUMBER = {1661157}, +MRREVIEWER = {M. Rajagopalan}, + DOI = {10.1215/kjm/1250518067}, + URL = {https://doi.org/10.1215/kjm/1250518067}, +} + +# Thomsen +@article {Thomsen91, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Nonstable {$K$}-theory for operator algebras}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {4}, + YEAR = {1991}, + NUMBER = {3}, + PAGES = {245--267}, + ISSN = {0920-3036}, + MRCLASS = {46L80 (19K33 46L85)}, + MRNUMBER = {1106955}, +MRREVIEWER = {Berndt Brenken}, + DOI = {10.1007/BF00569449}, + URL = {https://doi.org/10.1007/BF00569449}, +} + +@article {Thomsen93, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Finite sums and products of commutators in inductive limit {C}*-algebras}, + JOURNAL = {Ann. Inst. Fourier (Grenoble)}, + FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier}, + VOLUME = {43}, + YEAR = {1993}, + NUMBER = {1}, + PAGES = {225--249}, + ISSN = {0373-0956}, + MRCLASS = {46L05 (46M40)}, + MRNUMBER = {1209702}, +MRREVIEWER = {Guihua Gong}, + URL = {http://www.numdam.org/item?id=AIF_1993__43_1_225_0}, +} + +@article {Thomsen95, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Traces, unitary characters and crossed products by {$\mathbb{Z}$}}, + JOURNAL = {Publ. Res. Inst. Math. Sci.}, + FJOURNAL = {Kyoto University. Research Institute for Mathematical + Sciences. Publications}, + VOLUME = {31}, + YEAR = {1995}, + NUMBER = {6}, + PAGES = {1011--1029}, + ISSN = {0034-5318}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {1382564}, +MRREVIEWER = {Kevin McClanahan}, + DOI = {10.2977/prims/1195163594}, + URL = {https://doi.org/10.2977/prims/1195163594}, +} + + +@article {Thomsen97, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Limits of certain subhomogeneous {C}*-algebras}, + JOURNAL = {M\'{e}m. Soc. Math. Fr. (N.S.)}, + FJOURNAL = {M\'{e}moires de la Soci\'{e}t\'{e} Math\'{e}matique de France. Nouvelle S\'{e}rie}, + NUMBER = {71}, + YEAR = {1997}, + PAGES = {vi+125 pp. (1998)}, + ISSN = {0249-633X}, + MRCLASS = {46L05 (46L80 46M20)}, + MRNUMBER = {1649315}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.24033/msmf.385}, + URL = {https://doi.org/10.24033/msmf.385}, +} + +# Tikuisis +@article {Tikuisis12, + AUTHOR = {Tikuisis, Aaron}, + TITLE = {Regularity for stably projectionless, simple {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {263}, + YEAR = {2012}, + NUMBER = {5}, + PAGES = {1382--1407}, + ISSN = {0022-1236}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {2943734}, +MRREVIEWER = {Hiroyuki Osaka}, + DOI = {10.1016/j.jfa.2012.05.020}, + URL = {https://doi.org/10.1016/j.jfa.2012.05.020}, +} + +@article {Tikuisis16, + AUTHOR = {Tikuisis, Aaron}, + TITLE = {{K}-theoretic characterization of {C}*-algebras with approximately inner flip}, + JOURNAL = {Int. Math. Res. Not. IMRN}, + FJOURNAL = {International Mathematics Research Notices. IMRN}, + YEAR = {2016}, + NUMBER = {18}, + PAGES = {5670--5694}, + ISSN = {1073-7928}, + MRCLASS = {46L35 (19K99)}, + MRNUMBER = {3567256}, +MRREVIEWER = {Cristian Ivanescu}, + DOI = {10.1093/imrn/rnv334}, + URL = {https://doi.org/10.1093/imrn/rnv334}, +} + + +# Tikuisis-White-Winter +@article {TikuisisWhiteWinter17, + AUTHOR = {Tikuisis, Aaron and White, Stuart and Winter, Wilhelm}, + TITLE = {Quasidiagonality of nuclear {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {185}, + YEAR = {2017}, + NUMBER = {1}, + PAGES = {229--284}, + ISSN = {0003-486X}, + MRCLASS = {46L05 (47L40)}, + MRNUMBER = {3583354}, +MRREVIEWER = {Dinesh Jayantilal Karia}, + DOI = {10.4007/annals.2017.185.1.4}, + URL = {https://doi.org/10.4007/annals.2017.185.1.4}, +} + +# Toms +@article {Toms05, + AUTHOR = {Toms, Andrew S.}, + TITLE = {On the independence of {$K$}-theory and stable rank for simple + {C}*-algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {578}, + YEAR = {2005}, + PAGES = {185--199}, + ISSN = {0075-4102}, + MRCLASS = {46L80 (19K33 46L05)}, + MRNUMBER = {2113894}, +MRREVIEWER = {Nadia S. Larsen}, + DOI = {10.1515/crll.2005.2005.578.185}, + URL = {https://doi.org/10.1515/crll.2005.2005.578.185}, +} + +@article {Toms08, + AUTHOR = {Toms, Andrew S.}, + TITLE = {On the classification problem for nuclear {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {167}, + YEAR = {2008}, + NUMBER = {3}, + PAGES = {1029--1044}, + ISSN = {0003-486X}, + MRCLASS = {46L35 (19K14 46L80)}, + MRNUMBER = {2415391}, +MRREVIEWER = {Francesc Perera}, + DOI = {10.4007/annals.2008.167.1029}, + URL = {https://doi.org/10.4007/annals.2008.167.1029}, +} + +@article {Toms08b, + AUTHOR = {Toms, Andrew S.}, + TITLE = {An infinite family of non-isomorphic {C}*-algebras with + identical {$K$}-theory}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {360}, + YEAR = {2008}, + NUMBER = {10}, + PAGES = {5343--5354}, + ISSN = {0002-9947}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {2415076}, +MRREVIEWER = {Wilhelm Winter}, + DOI = {10.1090/S0002-9947-08-04583-2}, + URL = {https://doi.org/10.1090/S0002-9947-08-04583-2}, +} + +# Toms-Winter +@article {TomsWinter07, + AUTHOR = {Toms, Andrew S. and Winter, Wilhelm}, + TITLE = {Strongly self-absorbing {C}*-algebras}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {359}, + YEAR = {2007}, + NUMBER = {8}, + PAGES = {3999--4029}, + ISSN = {0002-9947}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {2302521}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.1090/S0002-9947-07-04173-6}, + URL = {https://doi.org/10.1090/S0002-9947-07-04173-6}, +} + +@article {TomsWinter08, + AUTHOR = {Toms, Andrew S. and Winter, Wilhelm}, + TITLE = {{$\mathcal{Z}$}-stable {ASH} algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {60}, + YEAR = {2008}, + NUMBER = {3}, + PAGES = {703--720}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (46L35 46L80)}, + MRNUMBER = {2414961}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.4153/CJM-2008-031-6}, + URL = {https://doi.org/10.4153/CJM-2008-031-6}, +} + +@article {TomsWinter09, + AUTHOR = {Toms, Andrew S. and Winter, Wilhelm}, + TITLE = {The {E}lliott conjecture for {V}illadsen algebras of the first + type}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {256}, + YEAR = {2009}, + NUMBER = {5}, + PAGES = {1311--1340}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {2490221}, +MRREVIEWER = {Xiao Chun Fang}, + DOI = {10.1016/j.jfa.2008.12.015}, + URL = {https://doi.org/10.1016/j.jfa.2008.12.015}, +} + +# Toms-White-Winer +@article {TomsWhiteWinter15, + AUTHOR = {Toms, Andrew S. and White, Stuart and Winter, Wilhelm}, + TITLE = {$\mathcal{Z}$-stability and finite-dimensional tracial + boundaries}, + JOURNAL = {Int. Math. Res. Not. IMRN}, + FJOURNAL = {International Mathematics Research Notices. IMRN}, + YEAR = {2015}, + NUMBER = {10}, + PAGES = {2702--2727}, + ISSN = {1073-7928}, + MRCLASS = {46L35 (46L40)}, + MRNUMBER = {3352253}, +MRREVIEWER = {C. J. K. Batty}, + DOI = {10.1093/imrn/rnu001}, + URL = {https://doi.org/10.1093/imrn/rnu001}, +} + +### UUUUUUUUUUUUUUUU + +### VVVVVVVVVVVVVVVVV +# Viladsen +@article {Villadsen98, + AUTHOR = {Villadsen, Jesper}, + TITLE = {Simple {C}*-algebras with perforation}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {154}, + YEAR = {1998}, + NUMBER = {1}, + PAGES = {110--116}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {1616504}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.1006/jfan.1997.3168}, + URL = {https://doi.org/10.1006/jfan.1997.3168}, +} + +@article {Villadsen99, + AUTHOR = {Villadsen, Jesper}, + TITLE = {On the stable rank of simple {C}*-algebras}, + JOURNAL = {J. Amer. Math. Soc.}, + FJOURNAL = {Journal of the American Mathematical Society}, + VOLUME = {12}, + YEAR = {1999}, + NUMBER = {4}, + PAGES = {1091--1102}, + ISSN = {0894-0347}, + MRCLASS = {46L05 (18B10 19K99 46L80 46M20)}, + MRNUMBER = {1691013}, +MRREVIEWER = {Vicumpriya S. Perera}, + DOI = {10.1090/S0894-0347-99-00314-8}, + URL = {https://doi.org/10.1090/S0894-0347-99-00314-8}, +} + +# Voiculescu +@article{Voiculescu83, + title={Asymptotically commuting finite rank unitary operators without commuting approximants}, + author={Voiculescu, Dan}, + journal={Acta Sci. Math.(Szeged)}, + volume={45}, + number={1-4}, + pages={429--431}, + year={1983} +} +@article {Voiculescu83, + AUTHOR = {Voiculescu, Dan}, + TITLE = {Asymptotically commuting finite rank unitary operators without + commuting approximants}, + JOURNAL = {Acta Sci. Math. (Szeged)}, + FJOURNAL = {Acta Universitatis Szegediensis. Acta Scientiarum + Mathematicarum}, + VOLUME = {45}, + YEAR = {1983}, + NUMBER = {1-4}, + PAGES = {429--431}, + ISSN = {0001-6969}, + MRCLASS = {47B44}, + MRNUMBER = {708811}, +} + +### WWWWWWWWWWWWWWWWWWWWWWWWWWWW +# Wiebel +@book {Kbook, + AUTHOR = {Weibel, Charles A.}, + TITLE = {The {$K$}-book}, + SERIES = {Graduate Studies in Mathematics}, + VOLUME = {145}, + NOTE = {An introduction to algebraic $K$-theory}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {2013}, + PAGES = {xii+618}, + ISBN = {978-0-8218-9132-2}, + MRCLASS = {19-01}, + MRNUMBER = {3076731}, +MRREVIEWER = {L. N. Vaserstein}, + DOI = {10.1090/gsm/145}, + URL = {https://doi.org/10.1090/gsm/145}, +} + +# Winter +@article {Winter11, + AUTHOR = {Winter, Wilhelm}, + TITLE = {Strongly self-absorbing {C}*-algebras are $\mathcal{Z}$-stable}, + JOURNAL = {J. Noncommut. Geom.}, + FJOURNAL = {Journal of Noncommutative Geometry}, + VOLUME = {5}, + YEAR = {2011}, + NUMBER = {2}, + PAGES = {253--264}, + ISSN = {1661-6952}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {2784504}, +MRREVIEWER = {Camillo Trapani}, + DOI = {10.4171/JNCG/74}, + URL = {https://doi.org/10.4171/JNCG/74}, +} + +@article {Winter12, + AUTHOR = {Winter, Wilhelm}, + TITLE = {Nuclear dimension and $\mathcal{Z}$-stability of pure {C}*-algebras}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {187}, + YEAR = {2012}, + NUMBER = {2}, + PAGES = {259--342}, + ISSN = {0020-9910}, + MRCLASS = {46L85 (46L35)}, + MRNUMBER = {2885621}, + DOI = {10.1007/s00222-011-0334-7}, + URL = {https://doi.org/10.1007/s00222-011-0334-7}, +} + +### XXXXXXXXXXXXXXXXXXXXX + +### YYYYYYYYYYYYYYYYYYYYY +@article {Yen56, + AUTHOR = {Yen, Ti}, + TITLE = {Isomorphism of unitary groups in {AW}*-algebras}, + JOURNAL = {Tohoku Math. J. (2)}, + FJOURNAL = {The Tohoku Mathematical Journal. Second Series}, + VOLUME = {8}, + YEAR = {1956}, + PAGES = {275--280}, + ISSN = {0040-8735}, + MRCLASS = {46.2X}, + MRNUMBER = {89376}, +MRREVIEWER = {I. E. Segal}, + DOI = {10.2748/tmj/1178244951}, + URL = {https://doi.org/10.2748/tmj/1178244951}, +} + +### ZZZZZZZZZZZZZZZZZZZZ + + diff --git a/Tensorially absorbing inclusions/letterfonts.tex b/Tensorially absorbing inclusions/letterfonts.tex new file mode 100644 index 0000000..89851e5 --- /dev/null +++ b/Tensorially absorbing inclusions/letterfonts.tex @@ -0,0 +1,77 @@ +%commands +\newcommand{\ms}{\mathscr} +\newcommand{\mc}{\mathcal} +\newcommand{\mf}{\mathfrak} +\newcommand{\nin}{\not\in} +\newcommand{\bs}{\backslash} +\newcommand{\nsg}{\unlhd} +\newcommand{\ov}{\overline} +%blackboard +\newcommand{\bN}{\mathbb{N}} +\newcommand{\bR}{\mathbb{R}} +\newcommand{\bZ}{\mathbb{Z}} +\newcommand{\bQ}{\mathbb{Q}} +\newcommand{\bF}{\mathbb{F}} +\newcommand{\bK}{\mathbb{K}} +\newcommand{\bG}{\mathbb{G}} +\newcommand{\bE}{\mathbb{E}} +\newcommand{\bC}{\mathbb{C}} +\newcommand{\bV}{\mathbb{V}} +\newcommand{\bA}{\mathbb{A}} +\newcommand{\bP}{\mathbb{P}} +\newcommand{\bD}{\mathbb{D}} +\newcommand{\bT}{\mathbb{T}} +\newcommand{\bM}{\mathbb{M}} +\newcommand{\bB}{\mathbb{B}} +\newcommand{\bL}{\mathbb{L}} +\newcommand{\bX}{\mathbb{X}} +\newcommand{\bY}{\mathbb{Y}} +\newcommand{\1}{{\bf 1}} +\newcommand{\0}{{\bf 0}} + +%mathcal +\newcommand{\cM}{\mathcal{M}} +\newcommand{\cN}{\mathcal{N}} +\newcommand{\cC}{\mathcal{C}} +\newcommand{\cB}{\mathcal{B}} +\newcommand{\cS}{\mathcal{S}} +\newcommand{\cI}{\mathcal{I}} +\newcommand{\cO}{\mathcal{O}} +\newcommand{\cP}{\mathcal{P}} +\newcommand{\cA}{\mathcal{A}} +\newcommand{\cL}{\mathcal{L}} +\newcommand{\cF}{\mathcal{F}} +\newcommand{\cH}{\mathcal{H}} +\newcommand{\cK}{\mathcal{K}} +\newcommand{\cD}{\mathcal{D}} +\newcommand{\cE}{\mathcal{E}} +\newcommand{\cT}{\mathcal{T}} +\newcommand{\cZ}{\mathcal{Z}} +\newcommand{\cU}{\mathcal{U}} +\newcommand{\cJ}{\mathcal{J}} +\newcommand{\cG}{\mathcal{G}} +\newcommand{\cR}{\mathcal{R}} +\newcommand{\cQ}{\mathcal{Q}} +\newcommand{\cY}{\mathcal{Y}} +%terns +\newcommand{\bh}{\mathcal{B}(\mathcal{H})} +\newcommand{\bk}{\mathcal{B}(\mathcal{K})} +\newcommand{\kh}{\mathcal{K}(\mathcal{H})} +\newcommand{\fh}{\mathcal{F}(\mathcal{H})} +\newcommand{\sa}{\mathcal{S}(\mathcal{A})} +\newcommand{\bx}{\mathcal{B}(\mathbb{X}} +\newcommand{\by}{\mathcal{C}(\mathbb{Y}} + +%frak +\newcommand{\fX}{\mathfrak{X}} +\newcommand{\fS}{\mathfrak{S}} +\newcommand{\fM}{\mathfrak{M}} +\newcommand{\fN}{\mathfrak{N}} +\newcommand{\fF}{\mathfrak{F}} +\newcommand{\fJ}{\mathfrak{J}} +\newcommand{\fT}{\mathfrak{T}} +\newcommand{\fA}{\mathfrak{A}} +\newcommand{\fB}{\mathfrak{B}} +\newcommand{\fP}{\mathfrak{P}} +\newcommand{\fQ}{\mathfrak{Q}} +\newcommand{\fO}{\mathfrak{O}} diff --git a/Tensorially absorbing inclusions/macros.tex b/Tensorially absorbing inclusions/macros.tex new file mode 100644 index 0000000..342d2ee --- /dev/null +++ b/Tensorially absorbing inclusions/macros.tex @@ -0,0 +1,104 @@ +% tensor products +\newcommand{\thk}{\cH\otimes\cK} +\newcommand{\amb}{\bA \otimes_{\max} \bB} +\newcommand{\omax}{\otimes_{\max}} +\newcommand{\omin}{\otimes_{\min}} +\newcommand{\ovn}{\overline{\otimes}} + +% probability +\newcommand{\PS}{(\Omega,\cM,\cP)} +\newcommand{\rvf}{f:\Omega \to \bR} +\newcommand{\linf}{\cL^{\infty -}\PS} +\newcommand{\probc}{\text{Prob}_c(\bR)} +\newcommand{\prob}{\text{Prob}(\bR)} + +% maps +\newcommand{\into}{\hookrightarrow} +\newcommand{\onto}{\twoheadrightarrow} +\newcommand{\act}{\curvearrowright} + +\newcommand{\ee}{\varepsilon} +\newcommand{\sm}{\setminus} +\newcommand{\Om}{\Omega} +\newcommand{\simi}{\sim_{\infty}} + +%limits +\newcommand{\limsupn}{\underset{n}{\text{limsup}}} +\newcommand{\sotc}{\stackrel{\text{SOT}}{\to}} +\newcommand{\wotc}{\stackrel{\text{WOT}}{\to}} +\newcommand{\sotlim}{\text{SOT-}\lim} +\newcommand{\wotlim}{\text{WOT-}\lim} + +% useful for nuclear, lifting maps, etc.... +\newcommand{\tphi}{\tilde{\phi}} +\newcommand{\tpsi}{\tilde{\psi}} +\newcommand{\hphi}{\hat{\phi}} +\newcommand{\hpsi}{\hat{\psi}} +\newcommand{\phil}{\phi_\lambda} +\newcommand{\psil}{\psi_\lambda} +\newcommand{\mkl}{M_{k(\lambda)}} +\newcommand{\mkn}{M_{k(n)}} +\newcommand{\phin}{\phi_n} +\newcommand{\psin}{\psi_n} +\newcommand{\vphi}{\varphi} + +\newcommand{\floor}[1]{\lfloor #1 \rfloor} + +\newcommand{\supp}{\text{supp}} +\newcommand{\wk}{\text{wk}} +\newcommand{\Tr}{\text{Tr}} +\newcommand{\dist}{\text{dist}} +\newcommand{\rank}{\text{rank}} +\newcommand{\sgn}{\text{sgn}} + +\newcommand{\ev}{\text{ev}} +\newcommand{\spr}{\text{spr}} +\newcommand{\tD}{\tilde{\Delta}} +\newcommand{\uv}{\underline} +\newcommand{\tr}{\text{tr}} +\newcommand{\id}{\text{id}} +\newcommand{\ad}{\text{ad}} +\newcommand{\Ad}{\text{Ad}} + +\newcommand*{\tc}[2]{(\ov{#1}^{#2},#2)} + +% Types for von Neumann algebras +\newcommand{\I}{\text{I}} +\newcommand{\II}{\text{II}} +\newcommand{\III}{\text{III}} +\newcommand{\IIIl}{\text{III}_{\lambda}} +% note the last character of this last one is an "ell", its just that my font has capital i and lower-case l being the same character..... + + +%operators +% basic algebra +\DeclareMathOperator{\Ann}{Ann} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\im}{im} + +% K-theory and regularity for C*-algebras +\DeclareMathOperator{\Ell}{Ell} +\DeclareMathOperator{\Aff}{Aff} +\DeclareMathOperator{\KT}{KT} +\DeclareMathOperator{\Cu}{Cu} +\DeclareMathOperator{\Ka}{K^{\text{alg}}} +\DeclareMathOperator{\Ku}{K^u} +\DeclareMathOperator{\hKa}{\ov{K}_1^{\text{alg}}} +\DeclareMathOperator{\hku}{\ov{K}_1^u} + +% exponential rank and length for C*-algebras +\newcommand{\cer}{\text{cer}} +\newcommand{\cel}{\text{cel}} + +% KK-theory and Ext +\DeclareMathOperator{\KK}{KK} +\DeclareMathOperator{\KKn}{KK_{\text{nuc}}} +\DeclareMathOperator{\KKs}{KK_{\text{sep}}} +\DeclareMathOperator{\Ext}{Ext} +\DeclareMathOperator{\Extn}{Ext_{\text{nuc}}} + +% direct/inverse limits +\newcommand{\dlim}{\underset{\to}{\lim}} +\newcommand{\ilim}{\underset{\leftarrow}{\lim}} diff --git a/Tensorially absorbing inclusions/preabmle.tex b/Tensorially absorbing inclusions/preabmle.tex new file mode 100644 index 0000000..767a975 --- /dev/null +++ b/Tensorially absorbing inclusions/preabmle.tex @@ -0,0 +1,52 @@ +% change this depending on what you're doing. +%\documentclass[11pt]{amsart} + +%packages +%\usepackage[margin=1.5in]{geometry} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[pdftex]{graphicx} +\usepackage{tikz-cd} +\usepackage{blindtext} + +\usepackage{hyperref,xcolor} +\usepackage{amsmath,amsfonts,amssymb,amsthm,color,mathtools,enumitem} +\usepackage{epstopdf} +\usepackage{polynom} +\usepackage{babel} +\usepackage{mathrsfs} +\usepackage{chngcntr} +\usepackage{verbatim} +\usepackage{hyphenat} + + +\hypersetup{ + colorlinks=true, + linkcolor=blue, + citecolor=cyan, + urlcolor=cyan} + + %theorems +\newtheorem{theorem}{Theorem}[section] +\newtheorem{defn}[theorem]{Definition} +\newtheorem{prop}[theorem]{Proposition} +\newtheorem{cor}[theorem]{Corollary} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{example}[theorem]{Example} +\newtheorem{remark}[theorem]{Remark} + +\newtheorem*{resultA}{Theorem A} +\newtheorem*{resultB}{Theorem B} +\newtheorem*{resultC}{Theorem C} +\newtheorem*{resultD}{Theorem D} +\newtheorem*{resultE}{Theorem E} +\newtheorem*{resultF}{Theorem F} + +%\newtheorem{result}{Theorem}[chapter] +\newtheorem{result}{Theorem} +%\renewcommand*{\theresult}{\arabic{chapter}.\Alph{result}} +\renewcommand*{\theresult}{\Alph{result}} +\newtheorem{resultcor}[result]{Corollary} + +\numberwithin{equation}{section} + diff --git a/Tensorially absorbing inclusions/tensorially_absorbing_inclusions.bbl b/Tensorially absorbing inclusions/tensorially_absorbing_inclusions.bbl new file mode 100644 index 0000000..66e3825 --- /dev/null +++ b/Tensorially absorbing inclusions/tensorially_absorbing_inclusions.bbl @@ -0,0 +1,288 @@ +\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} +\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } +% \MRhref is called by the amsart/book/proc definition of \MR. +\providecommand{\MRhref}[2]{% + \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} +} +\providecommand{\href}[2]{#2} +\begin{thebibliography}{BGSW22} + +\bibitem[AGJP22]{AGJP22} +Massoud Amini, Nasser Golestani, Saeid Jamali, and N.~Christopher Phillips, + \emph{Group actions on simple tracially $\mathcal{Z}$-absorbing + {C}*-algebras}, arXiv preprint arXiv:2204.03615 (2022). + +\bibitem[BGSW22]{BGSW22} +Joan Bosa, James Gabe, Aidan Sims, and Stuart White, \emph{The nuclear + dimension of $\mathcal{O}_\infty$-stable {C}*-algebras}, Adv. Math. + \textbf{401} (2022), Paper No. 108250, 51. \MR{4392219} + +\bibitem[Bis90]{Bisch90} +Dietmar~H. Bisch, \emph{On the existence of central sequences in subfactors}, + Trans. Amer. Math. Soc. \textbf{321} (1990), no.~1, 117--128. \MR{1005075} + +\bibitem[Bis94]{Bisch94} +\bysame, \emph{Central sequences in subfactors. {II}}, Proc. Amer. Math. Soc. + \textbf{121} (1994), no.~3, 725--731. \MR{1209417} + +\bibitem[BO08]{BrownOzawa} +Nathanial~P. Brown and Narutaka Ozawa, \emph{{C}*-algebras and + finite-dimensional approximations}, Graduate Studies in Mathematics, vol.~88, + American Mathematical Society, Providence, RI, 2008. \MR{2391387} + +\bibitem[BSKR93]{BSKR93} +Ola Bratteli, Erling St{\o}rmer, Akitaka Kishimoto, and Mikael R{\o}rdam, + \emph{The crossed product of a {UHF} algebra by a shift}, Ergodic Theory + Dynam. Systems \textbf{13} (1993), no.~4, 615--626. \MR{1257025} + +\bibitem[Bur10]{Burstein10} +Richard~D. Burstein, \emph{Commuting square subfactors and central sequences}, + Internat. J. Math. \textbf{21} (2010), no.~1, 117--131. \MR{2642989} + +\bibitem[Con75]{Connes75} +Alain Connes, \emph{Classification of automorphisms of hyperfinite factors of + type {II}$_1$ and {II}$_{\infty}$ and application to type {III} factors}, + Bull. Amer. Math. Soc. \textbf{81} (1975), no.~6, 1090--1092. \MR{388117} + +\bibitem[Con76]{Connes76} +\bysame, \emph{Classification of injective factors cases {II}$_1$, + {II}$_\infty$, {III}$_\lambda$, $\lambda \neq 1$}, Ann. of Math. (2) + \textbf{104} (1976), no.~1, 73--115. \MR{454659} + +\bibitem[CS19]{CameronSmith19} +Jan Cameron and Roger~R. Smith, \emph{A {G}alois correspondence for reduced + crossed products of simple {C}*-algebras by discrete groups}, Canad. J. Math. + \textbf{71} (2019), no.~5, 1103--1125. \MR{4010423} + +\bibitem[Cun77]{Cuntz77} +Joachim Cuntz, \emph{Simple {C}*-algebras generated by isometries}, Comm. 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Soc. + \textbf{12} (1999), no.~4, 1091--1102. \MR{1691013} + +\end{thebibliography} diff --git a/Tensorially absorbing inclusions/tensorially_absorbing_inclusions.tex b/Tensorially absorbing inclusions/tensorially_absorbing_inclusions.tex new file mode 100644 index 0000000..86d565c --- /dev/null +++ b/Tensorially absorbing inclusions/tensorially_absorbing_inclusions.tex @@ -0,0 +1,1207 @@ +\documentclass[12pt]{amsart} + +\usepackage[margin=1.5in]{geometry} + +\include{preabmle} +\include{macros} +\include{letterfonts} + +\begin{document} +\title{Tensorially absorbing inclusions of C*-algebras} +\author{Pawel Sarkowicz} +\email{\href{mailto:psark007@uottawa.ca}{psark007@uottawa.ca}} +\address{Department of Mathematics and Statistics, University of Ottawa, 75 Laurier Ave. East, Ottawa, ON, K1N 6N5 Canada} + + \begin{abstract} + When $\mathcal{D}$ is strongly self-absorbing we say an inclusion $B \subseteq A$ is $\mathcal{D}$-stable if it is isomorphic to the inclusion $B \otimes \mathcal{D} \subseteq A \otimes \mathcal{D}$. We give ultrapower characterizations and show that if a unital inclusion is $\mathcal{D}$-stable, then $\mathcal{D}$-stability can be exhibited for countably many intermediate C*-algebras concurrently. We show that such embeddings between $\mathcal{D}$-stable C*-algebras are point-norm dense in the set of all embeddings, and that every embedding between $\mathcal{D}$-stable C*-algebras is approximately unitarily equivalent to a $\mathcal{D}$-stable embedding. Examples are provided. + \end{abstract} + + \maketitle + \tableofcontents + + +\section{Introduction} + The study of inclusions of C*-algebras has been of recent interest. There is no short supply of research concerning inclusions relating to non-commutative dynamics \cite{Popa00,Izumi02,CameronSmith19,OsakaTeruya18,EchterhoffRordam21}, as well as inclusions of simple C*-algebras \cite{Rordam21}. We discuss inclusions from the lens of tensorially absorbing a strongly self-absorbing C*-algebra $\cD$ \cite{TomsWinter07}. + + When speaking of tensorial absorption with a strongly self-absorbing C*-algebra, central sequences play an imperative role akin to McDuff's character\hyp{}ization of when a $\II_1$ von Neumann algebra absorbs the unique hyperfinite $\II_1$ factor $\cR$ \cite{McDuff69}. Central sequences have been studied since the inception of operator algebras as Murray and von Neumann used them to exhibit non-isomorphic $\II_1$ factors by showing that $\cL(\bF_2)$ does not have property $\Gamma$ \cite{MvNIV}. They were also used in Connes' theorem concerning the uniqueness of $\cR$ \cite{Connes76}, and the classification of automorphisms on hyperfinite factors \cite{Connes75,Connes76}. In \cite{Bisch90,Bisch94}, Bisch considered the central sequence algebra $\cN^\omega \cap \cM'$ associated to an (irreducible) inclusion of $\II_1$ factors $\cN \subseteq \cM$ and characterized when there was an isomorphism $\Phi: \cM \simeq \cM \ov{\otimes} \cR$ such that $\Phi(\cN) = \cN \ov{\otimes} \cR$ in terms of there being non-commuting sequences in $\cN$ which asymptotically commute with the larger von Neumann algebra $\cM$ (in the $\|\cdot\|_2$-norm). As pointed out by Izumi \cite{Izumi04}, there are similar central characterizations for unital inclusions of separable C*-algebras which tensorially absorb a strongly self-absorbing C*-algebra $\cD$ (it was at least pointed out for $\cD$ being one of $M_{n^{\infty}},\cO_2,\cO_{\infty}$). + + For a strongly self-absorbing C*-algebra $\cD$ \cite[Definition 1.3(iv)]{TomsWinter07}, we study $\cD$-stable + inclusions (see Section \ref{section:dse-embeddings} for detailed definitions), analogous to Bisch's notion for an (irreducible) inclusion of $\II_1$ factors \cite{Bisch90}. We say that an inclusion $B \subseteq A$ is $\cD$-stable if there is an isomorphism $A \simeq A \otimes \cD$ such that +\begin{equation} + \begin{tikzcd} +A \arrow[r, "\simeq"] & A \otimes \cD \\ +B \arrow[r, "\simeq"] \arrow[u, hook, "\iota"] & B \otimes \cD \arrow[u, hook, "\iota \otimes \id_\cD"'] +\end{tikzcd} +\end{equation} +commutes. + +We study such inclusions systematically, discussing central sequence char\hyp{}acterizations, permanence properties, and giving examples towards the end. We list some key findings here. The first is that $\cD$-stable inclusions exist between $\cD$-stable C*-algebras if there is any inclusion, and that the set of $\cD$-stable inclusions is quite large. Moreover, as far as classification of embeddings up to approximate unitary equivalence (in particular by $K$-theory and traces), $\cD$-stable embeddings are all that matter. + +\begin{result}[Proposition \ref{allembeddingsareue}, Corollary \ref{pointnormdensity}] + Let $A,B$ be unital, separable, $\cD$-stable C*-algebras. + \begin{enumerate} + \item The set of $\cD$-stable embeddings $B \into A$ is point-norm dense in the set of all embeddings $B \into A$. + \item Every embedding $B \into A$ is approximately unitarily equivalent to a $\cD$-stable embedding. + \end{enumerate} + \end{result} + + We note that this set is however not everything. We provide examples of non-$\cD$-stable inclusions of $\cD$-stable C*-algebras, namely by fitting a C*-algebra with perforated Cuntz semigroup or with higher stable rank (in partic\hyp{}ular non-$\cZ$-stable C*-algebras) in between two $\cD$-stable C*-algebras. The second useful tool is that a $\cD$-stable inclusion allows one to find an appropriate isomorphism witnessing $\cD$-stability of countably many intermediate subalge\hyp{}bras at once. + + +\begin{result}[Theorem \ref{countablymanysubalgebras}] + Let $B \subseteq A$ be a unital, $\cD$-stable inclusion of separable C*-algebras. If $(C_n)$ is a sequence of C*-algebras such that $B \subseteq C_n \subseteq A$ unitally for all $n$, then there exists an isomorphism $\Phi: A \simeq A \otimes \cD$ such that + \begin{enumerate} + \item $\Phi(B) = B \otimes \cD$ and + \item $\Phi(C_n) = C_n \otimes \cD$ for all $n \in \bN$. + \end{enumerate} + \end{result} +This is not a trivial condition, as it is not true that any such isomorphism sends every intermediate C*-algebra to its tensor product with $\cD$ (see Example \ref{noteveryintermediate}). In fact, one can always find an intermediate C*-algebra $C$ between $B$ and $A$ and an isomorphism $A \simeq A \otimes \cD$ sending $B$ to $B \otimes \cD$ which does not send $C$ to $C \otimes \cD$ (although, of course, we will still have $C \simeq C \otimes \cD$). + +The above result, together with the Galois correspondence of Izumi \cite{Izumi02}, allows us to get a result similar to the main theorem of \cite{AGJP22}. There they prove that if $G \act^\alpha A$ is an action of a finite group with the weak tracial Rokhlin property on a C*-algebra $A$ with sufficient regularity conditions, then every C*-algebra between $A^\alpha \subseteq A$ and $A \subseteq A \rtimes_\alpha G$ is $\cZ$-stable. Assuming we have a unital C*-algebra with the same regularity conditions, we show that we can witness $\cZ$-stability of all such intermediate C*-algebras concurrently. + + + \begin{resultcor}[Corollary \ref{weaktracialRokhlinintermediate}] + Let $A$ be a unital, simple, separable, nuclear $\cZ$-stable C*-algebra and $G \act^\alpha A$ be an action of a finite group with the weak tracial Rokhlin property. There exists an isomorphism $\Phi: A \rtimes_\alpha G \simeq (A \rtimes_\alpha G) \otimes \cZ$ such that whenever $C$ is a unital C*-algebra satisfying either + \begin{enumerate} + \item $A^\alpha \subseteq C \subseteq A$ or + \item $A \subseteq C \subseteq A \rtimes_\alpha G$, + \end{enumerate} + we have $\Phi(C) = C \otimes \cZ$. + \end{resultcor} + + This paper is structured as follows. We discuss various local properties in Section \ref{section:dse-localembeddings}, and then formalize the notion of a $\cD$-stable embedding in Section \ref{section:dse-embeddings}, examining several properties and consequences. In Section \ref{section:dse-crossed-products} we show how several examples arising from non-commutative dynamical systems fit into the framework of $\cD$-stable inclusions. We finish with several examples in Section \ref{section:dse-examples}. + + %Acknowledgements section +% \addtocontents{toc}{\protect\setcounter{tocdepth}{0}} +% \section*{Acknowledgements} +% Many thanks to my supervisors Thierry Giordano and Aaron Tikuisis for many helpful discussions. + + %go back to normal TOC ordering.. +% \addtocontents{toc}{\protect\setcounter{tocdepth}{1}} + + %Acknowledgements section + \addtocontents{toc}{\protect\setcounter{tocdepth}{0}} + \section*{Acknowledgements} + Many thanks to my supervisors Thierry Giordano and Aaron Tikuisis for many helpful discussions, as well as to Eusebio Gardella for many helpful comments. + + %go back to normal TOC ordering.. + \addtocontents{toc}{\protect\setcounter{tocdepth}{1}} + + %\renewcommand*{\thetheorem}{\arabic{section}.\arabic{theorem}} + + + \section{Preliminaries}\label{preliminaries} + + \subsection{Notation} We will use capital letters $A,B,C,D$ to denote C*-algebras and usually a calligraphic $\cD$ to denote a strongly self-absorbing C*-algebra. Generally small letters $a,b,c,d,\dots,x,y,z$ will denote operators in C*-algebras. $A_+$ will denote cone of positive elements in a C*-algebra $A$. If $\ee > 0$ and $a,b$ are elements in a C*-algebra, we will write +\begin{equation} + a \approx_\ee b +\end{equation} + to mean that $\|a - b\| < \ee$. This will make some approximations more legible. + +% do we need the algebraic tensor product? So far, no. + The symbol $\otimes$ will denote the minimal tensor product of C*-algebras, while $\odot$ will mean the algebraic tensor product. We use the minimal tensor product throughout, and it is common for us to deal with nuclear C*-algebras so there should not be any ambiguity. The symbol $\ov{\otimes}$ will denote the von Neumann tensor product. + + We will denote by $M_n$ the C*-algebra of $n\times n$ matrices, and $M_{n^{\infty}}$ the uniformly hyperfinite (UHF) C*-algebra associated to the supernatural number $n^{\infty}$. We will write $\cQ$ for the universal UHF algebra $\cQ = \bigotimes_{n \in \bN} M_n$. + + By $G \act^\alpha A$, we will mean that the (discrete) group $G$ acts on $A$ by automorphisms, i.e., $\alpha: G \to \Aut(A)$ is a homomorphism. $A \rtimes_{r,\alpha}G$ will denote the reduced crossed product, which we will just write as $A \rtimes_\alpha G$ if it is clear from context that the group is amenable and $A$ is nuclear (e.g., if $G$ is finite). We will denote by $A^\alpha$ the fixed point subalgebra of the action (or $A^G$ if the action is clear from context). + + +\subsection{Ultrapowers, central sequences and central sequence algebras}\label{section:ultrapowers} + +Fix a free ultrafilter $\omega \in \beta \bN$. Throughout we will use ultrapowers to describe asymptotic behaviour, rather than sequences algebras. This comes down to a matter of taste and one can swap between the two if one so desires, as we will provide local characterizations as well. This also means that all of what we do will be independent of the specific ultrafilter $\omega$. + +For a C*-algebra $A$, the ultrapower of $A$ is the C*-algebra + \begin{equation} + A_\omega := \ell^{\infty}(A)/c_{0,\omega}(A), + \end{equation} +where $c_{0,\omega} := \{(a_n) \in \ell^{\infty}(A) \mid \lim_{n \to \omega} \|a_n\| = 0\}$ is the ideal of $\omega$-null sequences. We can embed $A$ into $A_\omega$ canonically by means of constant sequences: we identify $a \in A$ with the equivalence class of the constant sequence $(a)_n$. + +To ease notation, we will usually write elements of $A_\omega$ as sequences $(a_n)$, keeping in mind that these are equivalence classes without explicitly stating it every time. We note that the norm on $A_\omega$ is given by $\|(a_n)\| = \lim_{n \to \omega} \|a_n\|$. + +Kirchberg's $\ee$-test (\cite{Kirchberg06}, Lemma A.1) is essentially the operator algebraists' {\L}o{\' s}' theorem without having to turn to (continuous) model theory. Heuristically, it says that if certain things can be done approximately in an ultrapower, then certain things can be done exactly in an ultrapower. + +\begin{lemma}[Kirchberg's $\ee$-test]\label{lemma:Kirchberg-eptest} +Let $(X_n)_n$ be a sequence of sets and suppose that for each $n$, there is a sequence $(f_n^{(k)})$ of functions $f_n^{(k)}: X_n \to [0,\infty)$. For $k \in \bN$, let +\begin{equation} +f_\omega^k(s_1,s_2,\dots) := \lim_{n \to \omega} f_n^{(k)}(s_n). +\end{equation} +Suppose that for every $m \in \bN$ and $\ee > 0$, there is $s \in \prod_n X_n$ with $f_\omega^{(k)}(s) < \ee$ for $k=1,\dots,m$. Then there exists $t \in \prod_nX_n$ with $f_\omega^{(k)}(t) = 0$ for all $k \in \bN$. +\begin{comment} +Moreover, there is a sequence $n_1 < n_2 < \cdots$ in $\bN$ such that there are $s_l \in X_{n_l}$ with $f_{n_l}^{(k)} < 2^{-l}$ for $k \leq l, l \in \bN$. \end{comment} +\end{lemma} + +The above is useful, although if one so wishes, one can usually construct exact objects from approximate objects by using standard diagonalization arguments (under some separability assumptions). +These sorts of arguments work in both the ultrapower setting and the sequence algebra setting. + +Finally, if $\alpha \in \Aut(A)$ is an automorphism, there is an induced automorphism on $A_\omega$, which we will denote by $\alpha_\omega$ given by +\begin{equation} +\alpha_\omega((a_n)) := (\alpha(a_n)). +\end{equation} + + \subsection{Central sequences and central sequence subalgebras} For a unital C*-algebra $A$, the C*-algebra of $\omega$-central sequences is +\begin{equation} + A_\omega \cap A' = \{x \in A_\omega \mid [x,a] = 0 \text{ for all } a \in A\}, +\end{equation} + where we are identifying $A \subseteq A_\omega$ with the constant sequences. If $B \subseteq A$ is a unital C*-subalgebra and $S \subseteq A_\omega$ is a subset, we can associate the relative commutant of $S$ in $B_\omega$: +\begin{equation} + B_\omega \cap S' = \{b \in B_\omega \mid [b,s] = 0 \text{ for all } s \in S\}. +\end{equation} + Of particular interest will be when $S = A$, and $B \subseteq A$ is a unital inclusion of separable C*-algebras. + +\subsection{Strongly self-absorbing C*-algebras}\label{section:ssa-C*-algebras} + +A unital separable C*-algebra $\cD$ is strongly self-absorbing if $\cD \not\simeq \bC$ and there is an isomorphism $\phi: \cD \to \cD \otimes \cD$ which is approximately unitarily equivalent to the first factor embedding $d \mapsto d \otimes 1_\cD$ (see \cite{TomsWinter07}). All known strongly self-absorbing C*-algebras are: the Jiang-Su algebra $\cZ$ \cite{JiangSu99}, the Cuntz algebras $\cO_2$ and $\cO_{\infty}$ \cite{Cuntz77}, UHF algebras of infinite type, and $\cO_{\infty}$ tensor a UHF algebra of infinite type. Strongly self-absorbing C*-algebras have approximately inner flip, and therefore there are $K$-theoretic restrictions on strongly self-absorbing C*-alge\hyp{}bras -- see \cite{Tikuisis16,EndersSchemaitatTikuisis23}. + +Tensorial absorption with strongly self-absorbing C*-algebras gives rise to many regular properties, for examples in terms of $K$-theory, traces, and the Cuntz semigroup \cite{JiangSu99,Rordam91,Rordam92,Rordam04}. Of paramount interest is the Jiang-Su algebra $\cZ$. A cumulation of work has successfully classified all (unital) separable, simple, nuclear, infinite-dimensional, $\cZ$-stable C*-algebras satisfying the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet \cite{RosenbergSchochet87} by means of $K$-theory and traces. We describe how one might work with $\cZ$-stability in terms of its standard building blocks. Recall that, for $n,m \geq 2$, the dimension drop algebras are + \begin{equation} +\cZ_{n,m} := \{f \in C([0,1],M_n \otimes M_m) \mid f(0) \in M_n \otimes 1_{M_m}, f(1) \in 1_{M_n} \otimes M_m\}. + \end{equation} +Such an algebra is a called a prime dimension drop algebra when $n$ and $m$ are coprime. The Jiang-Su algebra $\cZ$ is the unique separable simple C*-algebra with unique tracial state which is an inductive limit of prime dimension drop algebras with unital connecting maps \cite{JiangSu99} (in fact, the dimension drop algebras can be chosen to have the form $\cZ_{n,n+1}$).It is $KK$-equivalent to $\bC$ and $\cZ$-stability is a necessary condition for $K$-theoretic classification. + +By \cite[Proposition 5.1]{RordamWinter10} (or \cite[Proposition 2.1]{Sato10} for our desired formulation), $\cZ_{n,n+1}$ is the universal C*-algebra generated by elements $c_1,\dots,c_n$ and $s$ such that +\begin{itemize} +\item $c_1 \geq 0$; +\item $c_ic_j^* = \delta_{ij}c_1^2$; +\item $s^*s + \sum_{i=1}^n c_i^*c_i = 1$; +\item $c_1s = s$. +\end{itemize} +If there are uniformly tracially large (in the sense of \cite[Definition 2.2]{TomsWhiteWinter15}) order zero\footnote{order zero meaning orthogonality preserving: $\phi: A \to B$ is c.p.c. order zero if it is c.p.c. and $\phi(a)\phi(b) = 0$ whenever $ab = 0$.} c.p.c. maps $M_n \to A_\omega \cap A'$, these give rise to elements $c_1,\dots,c_n \in A_\omega \cap A'$ with $c_1 \geq 0$ and $c_ic_j^* = \delta_{ij}c_1^2$, along with certain tracial information. If $A$ has strict comparison, Matui and Sato used this tracial information to show that $A$ has property (SI) \cite{MatuiSato12}, from which one can get an element $s \in A_\omega \cap A'$ such that $s^*s + \sum_{i=1}^n c_i^*c_i = 1$ and $c_1s = s$. This gives a *-homomorphism $\cZ_{n,n+1} \to A_\omega \cap A'$, which if can be done for each $n \in \bN$, is enough to conclude that $\cZ \into A_\omega \cap A'$ unitally and hence $A \simeq A \otimes \cZ$. In fact, it suffices to show that $\cZ_{2,3} \into A_\omega \cap A'$ (or $\cZ_{n,n+1}$ for some $n \geq 2$), see \cite[Theorem 3.4(ii)]{RordamWinter10} and \cite[Theorem 5.15]{Schemaitat22}. + +\section{Approximately central approximate embeddings}\label{section:dse-localembeddings} + + Here we formalize some results on approximate embeddings. When $B \subseteq A$ is a unital inclusion of separable C*-algebras, this will yield local characterizations of nuclear subalgebras of $B_\omega \cap A'$, as defined in (\ref{eq:central-sequence-subalgebra}). + + \begin{defn} + Let $B \subseteq A$ be a unital inclusion of C*-algebras and let $D$ be a unital, simple, nuclear C*-algebra. Let $\cF \subseteq D, \cG \subseteq A$ be finite sets and $\ee > 0$. We say that a u.c.p. map $\phi: D \to B$ is an $(\cF,\ee)$-approximate embedding if + \begin{enumerate} + \item $\phi(cd) \approx_\ee \phi(c)\phi(d)$ for all $c,d \in \cF$. + \end{enumerate} + If $\phi$ additionally satisfies + \begin{enumerate} + \item[2.] $[\phi(c),a] \approx_\ee 0$ for all $c \in \cF$ and $a \in \cG$, + \end{enumerate} + then we say that $\phi$ is an $(\cF,\ee,\cG)$-approximately central approximate embedding. + \end{defn} + + We will usually write that $\phi$ is a $(\cF,\ee)$-embedding or $(\cF,\ee,\cG)$-embedding to mean that $\phi$ is an $(\cF,\ee)$-approximate embedding or $(\cF,\ee,\cG)$-approximately central approximate embedding respectively. + + \begin{remark} + One can make a similar definition to the above if $D$ is not simple or nuclear (or even unital). The aim is to discuss subalgebras of $B_\omega \cap A'$, and if $D \into B_\omega \cap A'$ is nuclear, then one can use Choi-Effros to lift the embedding to a sequence of u.c.p. maps which are approximatey isometric, approximately multiplicative, and approximately commute with finite subsets of $A$. If $D$ is simple, the approximate isometry condition follows for free since the embedding $D \into B_\omega \cap A'$ must be isometric. + +If we loosen the simple and nuclear assumptions on $D$, we can still speak of bounded linear maps $\phi:D \to B$ (no longer necessarily u.c.p.) which are approximately isometric, approximately multiplicative, approximately adjoint-preserving, and approximately commute with a finite prescribed subset of $A$. This will allow one to discuss general subalgebras of $B_\omega \cap A'$. As we will only be interested in strongly self-absorbing subalgebras of $B_\omega \cap A'$, which are unital, separable, simple, and nuclear (see \cite[Section 1.6]{TomsWinter07}), we restrict ourselves to u.c.p. maps from a unital, simple, nuclear C*-algebras which are approximately multiplicative and approximately commute with finite subsets of $A$. + +Most of the work in this section can be done without assumptions of simplicity and nuclearity. + \end{remark} + + \begin{lemma}\label{lem:local-equiv-global} + Suppose that $A,B,D$ are unital C*-algebras with $B$ separable and $D$ simple, separable and nuclear. Let $S \subseteq A$ be a separable subset. There are $(\cF,\ee,\cG)$-approximately central approximate embeddings $D \to B$ for all $\cF \subseteq D, \cG \subseteq S$ and $\ee > 0$ if and only if there is a unital embedding $D \into B_\omega \cap S'$. + \begin{proof} + Let $(F_n)$ be an increasing sequence of finite subsets of $D$ with dense union and let $(G_n)$ be an increasing sequence of finite subsets of $S$ with dense union. Let $\phi_n: D \to B$ be $(F_n,\frac{1}{n},G_n)$-approximately central approximate embeddings. Let $\pi: \ell^{\infty}(B) \to B$ denote the quotient map and set + \begin{equation} + \psi := \pi \circ (\phi_n): D \to B_\omega +\end{equation} + which is a unital embedding such that $[\psi(d),a] = 0$ for all $d \in D$ and $a \in S$. + + For the other direction, suppose that $\psi: D \to B_\omega \cap S'$ is a unital embedding, $\cF \subseteq D,\cG \subseteq S$ are finite and $\ee >0$. By the Choi-Effros lifting theorem (see, for example, \cite[Theorem C.3]{BrownOzawa}) there is a u.c.p. lift $\tilde{\psi} = (\tilde{\psi}_n): D \to \ell^{\infty}(B)$ such that + \begin{itemize} + \item $\|\tilde{\psi}_n(cd) - \tilde{\psi}_n(c)\tilde{\psi}_n(d)\| \to^{n \to \omega} 0$, + \item $\|[\tilde{\psi}_n(d),a]\| \to^{n \to \omega} 0$ + \end{itemize} + for all $c,d \in D$ and $a \in A$. Take $n$ large enough and set $\phi = \psi_n$, so that $\phi$ will be a $(\cF,\ee,\cG)$-approximately central approximate embedding. + \end{proof} + \end{lemma} + + \begin{cor} + Let $A,B,D$ be unital C*-algebras with $B,D$ separable, simple and nuclear. Suppose that there are unital embeddings $\phi: D \to B_\omega$ and $\psi: B \to A_\omega$. Then there is a unital embedding $\xi: D \into A_\omega$. If $S \subseteq A_\omega$ is a separable subset with $\psi(B) \subseteq A_\omega \cap S'$, then $\xi$ can be chosen with $\xi(C) \subseteq A_\omega \cap S'$. + \begin{proof} + Let $\cF \subseteq D$ be finite and $\ee > 0$. Let $L := \max\{\max_{d \in \cF}\|d\|,1\}$. By the above lemma, there is an $(\cF,\frac{\ee}{2L})$-approximate embedding $\phi: D \to B$, so let $\cF' = \phi(\cF)$. Now there is an $(\cF',\frac{\ee}{2L})$-approximate embedding $\psi: B \to A$. An easy calculation shows that $\psi \circ \phi: D \to A$ is an approximate $(\cF,\ee)$-embedding. + + Appending the condition that $\psi: B \to A_\omega \cap S'$, then, for any finite subset $\cG \subseteq S$, we can take $\psi: B \to A$ to be a $(\cF',\frac{\ee}{2L},\cG)$-approximately central approximate embedding. This gives that $\psi \circ \phi: D \to A$ is be a $(\cF,\ee,\cG)$-approximately central approximate embedding. + \end{proof} + \end{cor} + + \begin{cor} + Let $D$ be a C*-algebra and $B \subseteq A$ be a unital inclusion of separable C*-algebras such that $B$ and $D$ are unital, separable, simple and nuclear. Suppose that there is an embedding $\pi: A \into A_\omega \cap A'$ with $\pi(B) \subseteq B_\omega \cap A'$. If $D \into B_\omega$ unitally, then $D \into B_\omega \cap A'$ unitally. + \begin{proof} + As $D \into B_\omega$ and $B \into B_\omega \cap A' \subseteq A_\omega \cap A'$, the above yields $D \into B_\omega \cap A'$. + \end{proof} + \end{cor} + + +The following is useful for discussing $\cD$-stability for some inclusions of fixed point subalgebras by certain automorphisms on UHF algebras. In particular, the following will work for automorphisms on UHF algebras of product-type, as well as tensor permutations (of finite tensor powers of UHF algebras). + +\begin{cor}\label{autfixedpoint} + Let $A = \bigotimes_\bN B$ be an infinite tensor product of a unital, separable, nuclear C*-algebra $B$ and let $D$ be unital, separable, simple, and nuclear. Let $\lambda \in \End(A)$ be the Bernoulli shift $\lambda(a) = 1 \otimes a$. If $\sigma \in \Aut(A)$ is such that $\lambda \circ \sigma = \sigma \circ \lambda$, and $D \into (A^\sigma)_\omega$ unitally, then $D \into (A^\sigma)_\omega \cap A'$ unitally. + \begin{proof} + Note that $\pi = (\lambda^n)$ induces an embedding $A \into A_\omega \cap A'$. We just need to show that $\pi(A^\sigma) \subseteq (A^\sigma)_\omega \cap A'$. The hypothesis gives that $\lambda^n \circ \sigma = \sigma \circ \lambda^n$ for all $n$, hence $\pi(A^\sigma) \subseteq (A^\sigma)_\omega \cap A'$. The result now follows from the above. + \end{proof} + \end{cor} + + We note that if we have approximately central approximate embeddings $D \to B \subseteq A$, then we can also find approximately central approximate embedding $D \to u^*Bu \subseteq A$ for any $u \in U(A)$. In the separable setting, this just means $D \into B_\omega \cap A'$ implies that $D \into u^*B_\omega u \cap A'$ for any $u \in U(A)$. + + \begin{lemma}\label{aduaapproximateembeddings} + Let $B \subseteq A$ be a unital inclusion of C*-algebras and let $D$ be a unital, separable, simple, nuclear C*-algebra. Let $u \in U(A)$. If there are $(\cF,\ee,\cG)$-approximately central approximate embeddings $D \to B$ for all $\cF \subseteq D, \cG \subseteq A$ finite subsets and $\ee > 0$, then there are $(\cF,\ee,\cG)$-approximately central approximate embeddings $D \to u^*Bu \subseteq A$ for all $\cF,\ee,\cG$. + \begin{proof} + Let $\cF \subseteq D, \cG \subseteq A$ be finite and $\ee > 0$. Let $L = \max\{1,\max_{d \in \cF}\|d\|\}$ and $\phi: D \to B$ be a $(\cF,\frac{\ee}{3L},\cG \cup \{u\})$-approximately central approximate embedding. Then $\psi = \Ad_u \circ \phi: D \to u^*Bu$ will be an $(\cF,\ee,\cG)$-embedding. + \end{proof} + \end{lemma} + + We can also discuss existence of approximately central approximate embed\hyp{}dings in inductive limits (with injective connecting maps). This is an adaptation of \cite[Proposition 2.2]{TomsWinter08} to our setting. + + \begin{prop}\label{approximateembeddinginductivelimit} + Suppose that we have increasing sequences $(B_n)$ and $(A_n)$ of C*-algebras such that $B_n \subseteq A_n$ are unital inclusions. + \begin{comment} +\begin{equation} + \begin{tikzcd} +B_1 \arrow[d, hook] \arrow[r, hook] & B_2 \arrow[d, hook] \arrow[r] & \cdots \arrow[r] & B \arrow[d, hook] \\ +A_1 \arrow[r, hook] & A_2 \arrow[r] & \cdots \arrow[r] & A, +\end{tikzcd} +\end{equation} +\end{comment} +If $B = \ov{\cup_n B_n}, A = \ov{\cup_n A_n}$, and $D = \ov{\cup_nD_n}$ where $(D_n)$ is an increasing sequence of unital, separable, simple, nuclear C*-algebras and there are $(\cF,\ee,\cG)$-embeddings $D_n \to B_n \subseteq A_n$ whenever $n \in \bN, \cF \subseteq D_n, \cG \subseteq A_n$ are finite and $\ee > 0$, then there are $(\cF,\ee,\cG)$-embeddings $D \to B \subseteq A$ for all $\cF \subseteq D,\cG \subseteq A$ finite and $\ee > 0$. + \begin{proof} + Let $\cF \subseteq \cD$ and $\cG \subseteq A$ be finite sets and $\ee > 0$. Let +\begin{equation} + L := \max\{1,\max_{d \in \cF}\|d\|,\max_{a \in \cG}\|a\|\} +\end{equation} + and set $\delta := \frac{\ee}{6L+5}$. Without loss of generality assume that $\ee < 1$. Label $\cF = \{d_1,\dots,d_p\}$ and $\cG = \{a_1,\dots,a_q\}$ and find $N$ large enough so that there are $d_1',\dots,d_p' \in D_N$ and $a_1',\dots,a_q' \in A_N$ with $d_i' \approx_\delta d_i$ and $a_j' \approx_\delta a_j$. Let $\cF':=\{d_1',\dots,d_p'\}, \cG':=\{a_1',\dots,a_q'\}$ and let $\phi: D_N \to B_N \subseteq A_N$ be an $(\cF',\delta,\cG')$-embedding. As $D_N$ is nuclear, there are $k \in \bN$ and u.c.p. maps $\rho: D_N \to M_k$ and $\eta: M_k \to B_N$ such that $\eta \circ \rho(d_i') \approx_\delta \phi(d_i')$ and $\eta \circ \rho(d_i'd_j') \approx_\delta \phi(d_i'd_j')$. Use Arveson's extension theorem (see \cite[Section 1.6]{BrownOzawa}) to extend $\rho$ to a u.c.p. map $\tilde{\rho}: D \to M_k$ and let $\psi := \eta \circ \tilde{\rho}: D \to B_N$. As $B_N \subseteq B$, we can think of $\psi$ as a map $\psi: D \to B$. Now for $i=1,\dots,p$, we have +\begin{equation} +\begin{split} + \psi(d_id_j) &\approx_{(2L+1)\delta}\psi(d_i'd_j') \\ + &= \eta \circ \rho(d_i'd_j') \\ + &\approx_\delta \phi(d_i'd_j') \\ + &\approx_\delta \phi(d_i')\phi(d_j') \\ + &\approx_{2L\delta} \eta \circ \rho(d_i')\eta \circ \rho(d_j') \\ + &= \psi(d_i')\psi(d_j') \\ + &\approx_{(2L+1)\delta}\psi(d_i)\psi(d_j). +\end{split} +\end{equation} + Thus $\psi(d_id_j) \approx_{(4+6L)\delta} \psi(d_i)\psi(d_j)$, and as $(4 + 6L)\delta \leq (6L + 5)\delta = \ee$, this implies that $\psi(d_id_j) \approx_\ee \psi(d_i)\psi(d_j)$. For approximate commutation with $\cG$, we make use of the following two approximations: for $a,a',a'',b,b'$ elements in a C*-algebra, + \begin{equation} + \begin{split} + &\|[a,b]\| \leq (\|a\| + \|a'\|)\|b - b'\| + (\|b\| + \|b'\|)\|a - a'\| + \|[a',b']\|, \\ + &\|[a',b']\| \leq 2\|b'\|\|a' - a''\| + \|[a'',b']\|. + \end{split} + \end{equation} + Note that for $a = \psi(d_i),a' = \psi(d_i'),a'' = \phi(d_i'), b = a_j, b' = a_j'$, we have that $\|a\|,\|b\| \leq L + 1$ and $\|a'\|,\|a''\|,\|b'\| \leq L$. Therefore from the above two inequalities we get + \begin{equation} + \begin{split} + &\|[\psi(d_i),a_j]\| \leq 2L\|\psi(d_i) - \psi(c_i')\|+ 2(L+1)\|a_j - a_j'\| + \|[\psi(d_i'),a_j]\|; \\ + &\|[\psi(d_i'),a_j']\| \leq 2(L+1)\|\psi(d_i') - \phi(d_i')\| + \|[\phi(d_i'),a_j']\|. + \end{split} + \end{equation} + Using these approximations we have +\begin{equation} +\begin{split} + \|[\psi(d_i),a_j]\| &\leq 2L\|\psi(d_i) - \psi(d_i')\| + 2(L+1)\|a_j - a_j'\| + \|[\psi(d_i'),a_j]\| \\ + &< (4L+2)\delta + \|[\psi(d_i'),a_j]\| \\ + &\leq (4L+2)\delta + 2(L+1)\|\psi(d_i') - \phi(d_i')\| + \|[\phi(c_i'),a_j']\| \\ + &< (4L+2)\delta + 2(L+1)\delta + \delta \\ + &= (6L+5)\delta = \ee. +\end{split} +\end{equation} + \end{proof} + \end{prop} + +The following will be useful to show that there are many $\cD$-stable embeddings. + +\begin{lemma}\label{conjugations} + Let $\phi: B \simeq B'$ and $\psi: A \simeq A'$ be isomorphisms between unital C*-algebras and let $D$ be a unital, simple, nuclear C*-algebra. Suppose that there is a *-homomorphism $\eta: B' \into A'$ such that there are $(\cF,\ee,\cG)$-embeddings $D \to \eta(B') \subseteq A'$ for all finite subsets $\cF \subseteq D, \cG \subseteq A'$ and $\ee > 0$. Let $\sigma = \psi^{-1} \circ \eta \circ \phi: B \to A$. Then there are $(\cF,\ee,\cG)$-embeddings $D \to \sigma(B) \subseteq A$ for all $\cF \subseteq D, \cG \subseteq A$ finite and $\ee > 0$. + \begin{proof} + The diagram + \begin{equation} + \begin{tikzcd} +A \arrow[r, "\psi"] & A' \\ +B \arrow[u, "\sigma"] \arrow[r, "\phi"'] & B' \arrow[u, "\eta"'] +\end{tikzcd} + \end{equation} + commutes, and so if $\cF \subseteq D, \cG \subseteq A$ are finite, $\ee > 0$ and $\xi: D \to \eta(B') \subseteq A'$ is an $(\cF,\ee,\psi(\cG))$-embedding, then $\psi^{-1} \circ \xi: D \to \psi^{-1}(\eta(B')) \subseteq \psi^{-1}(A') = A$ is an $(\cF,\ee,\cG)$ embedding. Moreover, from + \begin{equation} + \psi^{-1}(\eta(B')) = \psi^{-1}(\eta(\phi(B))) = \sigma(B), + \end{equation} + its clear that $\psi^{-1} \circ \xi$ is an $(\cF,\ee,\cG)$ embedding $D \to \sigma(B) \subseteq A$. + \begin{comment} + Let $\cF \subseteq D, \cG \subseteq A$ be finite and $\ee > 0$. Let $\cG' = \psi(\cG)$, and $\tilde{\xi}: D \to \eta(B')$ be an $(\cF,\ee,\cG')$-embedding, and lift it to a u.c.p. map $\xi: D \to B$ by Choi-Effros. We claim that $\sigma \circ \phi^{-1} \circ \xi: D \to \sigma(B)$ is a $(\cF,\ee,\cG)$-embedding $D \to \sigma(B) \subseteq A$. We have +\begin{equation} + \sigma \circ \phi^{-1} \circ \xi = \psi^{-1} \circ \eta \circ \phi \circ \phi^{-1} \circ \xi = \psi^{-1} \circ \eta \circ \xi = \psi^{-1} \circ \tilde{\xi}. +\end{equation} + As $\xi(cd) \approx_\ee \xi(c)\xi(d)$ for $c,d \in \cF$, it is clear that $\psi^{-1} \circ \xi$ will be $(\cF,\ee)$-multiplicative, and consequently so will $\sigma \circ \phi^{-1} \circ \xi$. For approximate commutation, we have for $d \in \cF$ and $a \in \cG$: +\begin{equation} +\begin{split} + \|[\sigma(\phi^{-1} \circ \xi(d)),a]\| &= \|[\psi^{-1} \circ \eta \circ \phi \circ \phi^{-1} \circ \xi(d),a]\| \\ + &= \|[\psi^{-1} \circ \eta \circ \xi(d),a]\| \\ + &= \|[\psi \circ \psi^{-1} \circ \eta \circ \xi(d),\psi(a)]\| \\ + &= \|[\eta \circ \xi(d),\psi(a)]\| \\ + &= \|[\tilde{\xi}(d),\psi(a)]\| \\ + &< \ee +\end{split} +\end{equation} + where the third equality follows since $\psi$ is isometric. + \end{comment} + \end{proof} + \end{lemma} + + \section{Relative intertwinings and $\cD$-stable embeddings}\label{section:dse-embeddings} + + \subsection{Relative intertwinings}\label{ssectoin:dse-relative-intertwinings} It is well known that a strongly self-absorbing C*-algebra $\cD$ embeds unitally into the central sequence algebra $(\cM(A))_\omega \cap A'$ of a separable C*-algebra $A$ if and only if $A \simeq A \otimes \cD$, where $\cM(A)$ is the multiplier algebra of $A$ (for example, \cite[Theorem 7.2.2(i)]{RordamBook}). We alter the proof to keep track of a subalgebra in order to show that for a unital inclusion $B \subseteq A$ of separable C*-algebras, $\cD \into B_\omega \cap A'$ if and only if there is an isomorphism $\Phi: A \to A \otimes \cD$, which is approximately unitarily equivalent to the first factor embedding, and satisfies $\Phi(B) = B \otimes \cD$. This was initially done for (irreducible) inclusions of $\II_1$ factors in \cite{Bisch90} and commented on in \cite{Izumi04} for $\cD$ being $M_{n^{\infty}},\cO_2,\cO_{\infty}$. The proof we alter is Elliott's intertwining argument, which can be found as a combination of Proposition 2.3.5, Proposition 7.2.1 and Theorem 7.2.2 of \cite{RordamBook}. + + \begin{prop}[Relative intertwining]\label{relativeintertwining} + Let $A,B,C$ be unital, separable C*-algebras, and let $\phi: A \into C, \theta: B \to A, \psi: B \to C$ be unital *-homomorphisms such that $\phi \circ \theta(B) \subseteq \psi(B)$. Suppose there is a sequence$(u_n) \subseteq \psi(B)_\omega \cap \phi(A)'$ of unitaries such that + \begin{itemize} + \item $\dist(v_n^*cv_n,\phi(A)_\omega) \to 0$ for all $c \in C$; + \item $\dist(v_n^*\psi(b)v_n,\phi \circ \theta(B)_\omega) \to 0$ for all $b \in B$. + \end{itemize} + Then $\phi$ is approximately unitarily equivalent to an isomorphism $\Phi: A \simeq C$ such that $\Phi \circ \theta(B) = \psi(B)$. + \begin{proof} + Apply the below proposition with $B_m := B, \theta_m := \theta, \psi_m := \psi$ for all $m \in \bN$. + \end{proof} + \end{prop} + + \begin{prop}[Countable relative intertwining]\label{countablerelativeintertwining} + Let $A,B_m,C$ be unital, separable C*-algebras, $m \in \bN$, and $\phi: A \into C,\theta_m: B_m \to A,\psi_m: B_m \to C$ be such that $\phi \circ \theta_m(B_m) \subseteq \psi_m(B_m)$ and $\psi_1(B_1) \subseteq \psi_m(B_m)$. Suppose there is a sequence $(v_n) \subseteq \psi_1(B_1)_\omega \cap \phi(A)'$ of unitaries such that + \begin{itemize} + \item $\dist(v_n^*cv_n,\phi(A)_\omega) \to 0$ for all $c \in C$; + \item $\dist(v_n^*\psi_m(b)v_n,\phi \circ \theta_m(B_m)_\omega) \to 0$ for all $b \in B_m$. + \end{itemize} + Then $\phi$ is approximately unitarily equivalent to an isomorphism $\Phi:A \simeq C$ such that $\Phi \circ \theta_m (B_m) = \psi_m(B_m)$ for all $m \in \bN$. + \begin{proof} + We show that if there are unitaries $(v_n) \subseteq \psi_1(B_1)$ satisfying + \begin{itemize} + \item $[v_n,\phi(a)] \to 0$ for all $a \in A$; + \item $\dist(v_n^*cv_n,\phi(A)) \to 0$ for all $c \in C$; + \item $\dist(v_n^*\psi_m(b)v_n,\phi \circ \theta_m(B_m)) \to 0$ for all $b \in B_m$, + \end{itemize} + then the conclusion holds. Such unitaries can be found using Kirchberg's $\ee$-test (Lemma \ref{lemma:Kirchberg-eptest}). + + Let $(a_n),(b_n^{(m)}),(c_n)$ be dense sequences of $A,B_m,C$ respectively. We can inductively choose $v_n$, forming a subsequence $(v_n)$ of the unitaries above (after reindexing, we are still calling them $v_n$), such that there are $a_{jn} \in A, b_{jn}^{(m)} \in B_m$ with + \begin{itemize} + \item $v_n^*\cdots v_1^*c_jv_1\cdots v_n \approx_{\frac{1}{n}} \phi(a_{jn})$; + \item $v_n^*\cdots v_1^*\psi(b_j^{(m)})v_1\cdots v_n \approx_{\frac{1}{n}} \phi \circ \theta_m(b_{jn}^{(m)})$; + \item $[v_n,\phi(a_j)] \approx_{\frac{1}{2^n}} 0$; + \item $[v_n,\phi(a_{jl})] \approx_{\frac{1}{2^n}} 0$; + \item $[v_n,\phi \circ \theta_m(b_j^{(m)})] \approx_{\frac{1}{2^n}} 0$; + \item $[v_n,\phi \circ \theta_m(b_{jl}^{(m)})] \approx_{\frac{1}{2^n}} 0$, + \end{itemize} + where $j,m=1,\dots,n$ and $l=1,\dots,n-1$. Define, for $a \in \{a_n \mid n \in \bN\}$, +\begin{equation} + \Phi(a) = \lim_n v_1\cdots v_n\phi(a) v_n^*\cdots v_1^* +\end{equation} + which extends to a *-isomorphism $\Phi: A \simeq C$, as in \cite[Proposition 2.3.5]{RordamBook}. The proof also yields the following useful approximation: +\begin{equation} + \Phi \circ \theta_m(b_{jn}^{(m)}) \approx_{\frac{1}{2^n}} v_1 \cdots v_n \phi \circ \theta_m(b_{jn}^{(m)})v_n^* \cdots v_1^* +\end{equation} + for appropriate $n \geq j,m$. + + We now need to check that $\Phi \circ \theta_m(B_m) = \psi_m(B_m)$. Approximate +\begin{equation} + \psi_m(b_j^{(m)}) \approx_{\frac{1}{n}} v_1\cdots v_n \phi \circ \theta_m(b_{jn}^{(m)})v_n^*\cdots v_1^* \approx_{\frac{1}{2^n}} \Phi \circ \theta_m(b_{jn}^{(m)}). +\end{equation} + This yields $\psi_m(B_m) \subseteq \ov{\Phi \circ \theta_m(B_m)} = \Phi \circ \theta_m(B_m)$. On the other hand for any $\ee > 0$ and $b \in B_m$, we can find $n$ such that +\begin{equation} + \Phi \circ \theta_m(b) \approx_\ee v_1\cdots v_n \phi \circ \theta_m(b)v_n^*\cdots v_1^* \in \psi_m(B_m) +\end{equation} + since $v_i \in \psi_1(B_1) \subseteq \psi_m(B_m)$ and $\phi \circ \theta_m(B_m) \subseteq \psi_m(B_m)$. Hence $\Phi \circ \theta_m(B_m) \subseteq \ov{\psi_m(B_m)} = \psi_m(B_m)$. + \end{proof} + \end{prop} + + \subsection{$\cD$-stable embeddings}\label{ssection:dse-d-stable-embeddings} + + \begin{defn} + Let $\iota: B \into A$ be an embedding and $\cD$ be strongly self-absorbing. We say that $\iota$ is $\cD$-stable (or $\cD$-absorbing) if there exists an isomorphism $\Phi:A \simeq A \otimes \cD$ such that $\Phi \circ \iota(B) = \iota(B) \otimes \cD$. + \end{defn} + + + We will mostly have interest in the case where $\iota$ corresponds to the inclusion map and $B \subseteq A$ is a subalgebra. In this form, we will say $B \subseteq A$ is $\cD$-stable (or $\cD$-absorbing). Clearly $\iota$ being $\cD$-stable is the same as $\iota(B) \subseteq A$ being $\cD$-stable. We note that we can define the above for any *-homomorphism. Namely, a *-homomorphism $\phi:B \to A$ is $\cD$-stable if $\phi(B) \subseteq A$ is. + + + + \begin{lemma}\label{trivialinclusion} + If $\iota: B \into A$ is an embedding, then $\iota \otimes \id_D: B \otimes \cD \into A \otimes \cD$ is $\cD$-stable. + \begin{proof} + Let $\phi: D \simeq D \otimes \cD$ be an isomorphism. Then +\begin{equation} + \Phi := \id_A \otimes \phi: A \otimes \cD \to A \otimes \cD \otimes \cD +\end{equation} + is an isomorphism with +\begin{equation} + \Phi(\iota \otimes \id_\cD(B \otimes \cD)) = \left(\iota \otimes \id_\cD(B \otimes \cD)\right) \otimes \cD. +\end{equation} + \end{proof} + \end{lemma} + + We note that this is a strengthening of the notion of $\cD$-stability for C*-algebras because if $\iota = \id_A: A \to A$, then $\iota$ is $\cD$-stable if and only if $A$ is $\cD$-stable. This condition is different than the notion of $\cO_2$ or $\cO_{\infty}$-absorbing morphisms discussed in \cite{BGSW22,Gabe20,Gabe19} -- they require sequences from a larger algebra to commute with a smaller algebra, while we require sequences from a smaller algebra to commute with the larger algebra. + + The following adapts \cite[Theorem 7.2.2]{RordamBook}. + \begin{theorem}\label{equivDstable} + Suppose that $B \subseteq A$ is a unital inclusion of separable C*-algebras. If $\cD$ is strongly self-absorbing, then $B \subseteq A$ is $\cD$-stable if and only if there is a unital inclusion $\cD \into B_\omega \cap A'$. + \begin{proof} + Let $\phi: A \to A \otimes \cD$ be the first factor embedding $\phi(a) := a \otimes 1_\cD$. First suppose that $\xi: \cD \into B_\omega \cap A' \simeq (B \otimes 1_\cD)_\omega \cap (A \otimes 1_\cD)'$ is an embedding (so that $\phi(a)\xi(d) \in \phi(A)_\omega$ and $\phi(b)\xi(d) \in \phi(B)_\omega$). Let $\eta: \cD \into (B \otimes \cD)_\omega \cap (A \otimes 1_\cD)'$ be given by $\eta(d) := (1 \otimes d)_n$ and notice that $\xi,\eta$ have commuting ranges. As all endomorphisms of $\cD$ are approximately unitarily equivalent by \cite[Corollary 1.12]{TomsWinter07}, let $(v_n) \subseteq C^*(\xi(\cD),\eta(\cD)) \simeq \cD \otimes \cD$ be such that $v_n^*\eta(d)v_n \to \xi(d)$ for $d \in \cD$. For $b \in B$ and $d \in \cD$, we have +\begin{equation} +\begin{split} + v_n^*(b \otimes d)v_n &= v_n^*(b \otimes 1_\cD)(1_A \otimes d)v_n^* \\ + &= v_n^*\phi(b)\eta(d)v_n \\ + &= \phi(b)v_n^*\eta(d)v_n \\ + &\to \phi(b)\xi(d) \in \phi(B)_\omega. + \end{split} +\end{equation} + Moreover the same argument shows that, for $a \in A$, we have +\begin{equation} + v_n^*(a \otimes d)v_n \to \phi(a)\xi(d) \in \phi(A)_\omega. +\end{equation} + Now $(v_n)$ satisfy the hypothesis of Proposition \ref{relativeintertwining} with $C := A \otimes D$, $\phi$ being the first factor embedding, $\theta: B \to A$ being the inclusion and $\psi: B \simeq B \otimes \cD \subseteq A \otimes \cD = C$ (where this isomorphism exists since if $\cD \into B_\omega \cap A'$, then clearly $\cD \into B_\omega \cap B'$). From this we see that $\phi$ is approximately unitarily equivalent to an isomorphism $\Phi:A \simeq A \otimes \cD$ such that $\Phi(B) = B \otimes \cD$. + + Conversely, if $B \subseteq A$ is $\cD$-stable, let $\Phi: A \simeq A \otimes \cD$ be an isomorphism such that $\Phi(B) = B \otimes \cD$. By \cite[Proposition 1.10(iv)]{TomsWinter07}, we can identify $\cD \simeq \cD^{\otimes \infty}$ and take $\xi: \cD \into B_\omega \cap A'$ to be given by +\begin{equation} + \xi(d) = (\Phi^{-1}(1_A \otimes 1_\cD^{\otimes n-1} \otimes d \otimes 1_\cD^{\otimes \infty}))_n. +\end{equation} + \end{proof} + \end{theorem} + + \begin{cor} + Let $\iota: B \into A$ be a unital embedding between separable C*-algebras. If $\cD$ is strongly self-absorbing and $\iota$ is $\cD$-stable, then for every intermediate unital C*-algebra $C$ with $\iota(B) \subseteq C \subseteq A$, we have that $\iota(B) \subseteq C$ and $C \subseteq A$ are $\cD$-stable. In particular, $C \simeq C \otimes \cD$ for all such $C$. + \begin{proof} + We have + \begin{equation} + \cD \into B_\omega \cap A' \subseteq B_\omega \cap C' + \end{equation} + and + \begin{equation} + \cD \into B_\omega \cap A' \subseteq C_\omega \cap A'. + \end{equation} + \end{proof} + \end{cor} + + It is not however the case that any isomorphism $\Phi: A \simeq A \otimes \cD$ with $\Phi(B) = B \otimes \cD$ maps $C$ to $C \otimes \cD$. + + \begin{example}\label{noteveryintermediate} + Let $\cD$ be strongly self-absorbing and consider +\begin{equation} +\begin{split} + A &:= \cD \otimes \cD \otimes \cD, \\ + C_1 &:= \cD \otimes 1_\cD \otimes \cD,\\ + C_2 &:= 1_\cD \otimes \cD \otimes \cD,\\ + B &:= 1_\cD \otimes 1_\cD \otimes \cD. +\end{split} +\end{equation} + If $f: \cD \otimes \cD \to \cD \otimes \cD$ is the tensor flip and $\phi: \cD \simeq \cD \otimes \cD$ is an isomorphism, let +\begin{equation} + \Phi := f \otimes \phi: A \simeq A \otimes \cD +\end{equation} + which satisfies $\Phi(B) = B \otimes \cD$ (in particular $B \subseteq A$ is $\cD$-stable). However, +\begin{equation} + \Phi(C_1) = C_2 \otimes \cD \text{ and } \Phi(C_2) = C_1 \otimes \cD. +\end{equation} + \end{example} + + In fact the above example can be generalized to show that for any unital $\cD$-stable inclusion $B \subseteq A$, there is an isomorphism $\Phi: A \simeq A \otimes \cD$ with $\Phi(B) = B \otimes \cD$, and some intermediate algebra $B \subseteq C \subseteq A$ with $\Phi(C) \neq C \otimes \cD$ (obviously we will always have that $\Phi(C) \simeq C \simeq C \otimes \cD$, but equality may not happen). + + \begin{cor} + Let $B \subseteq A$ be a $\cD$-stable inclusion. There exist a C*-algebra $C$ with $B \subseteq C \subseteq A$ and an isomorphism $\Phi: A \simeq A \otimes \cD$ such that $\Phi(B) = B \otimes \cD$ but $\Phi(C) \neq C \otimes \cD$. + \end{cor} + + + However, we can always realize $\cD$-stability for countably many intermediate C*-algebras at once using \emph{some} isomorphism $A \simeq A \otimes \cD$. + + \begin{theorem}\label{countablymanysubalgebras} + Suppose that $B_1 \subseteq B_m \subseteq A$ are unital inclusions of separable C*-algebras (note that we are {\bf not} asking for $(B_m)$ to form a chain). If $\cD$ is strongly self-absorbing and $\cD \into (B_1)_\omega \cap A'$ unitally, there exists an isomorphism $\Phi: A \simeq A \otimes \cD$ such that $\Phi(B_m) = B_m \otimes \cD$ for all $m \in \bN$. + \begin{proof} + This is essentially the same proof as Theorem \ref{equivDstable}, except we use the countable relative intertwining (Proposition \ref{countablerelativeintertwining}) in place of Proposition \ref{relativeintertwining}. Let $\xi,\eta$ be as before and let $(v_n) \subseteq C^*(\xi(\cD),\eta(\cD)) \simeq \cD \otimes \cD$ be such that $v_n^*\eta(d)v_n \to \xi(d)$ for $d \in \cD$. + \begin{itemize} + \item If $a \in A, d \in \cD, v_n^*(a \otimes d)v_n \to \phi(a)\xi(d) \in \phi(A)_\omega$; + \item if $b \in B_m, v_n^*(b \otimes d)v_n \to \phi(b)\xi(d) \in \phi(B_m)_\omega$. + \end{itemize} + Now with $\phi: A \to A \otimes \cD$ the first factor embedding, $\theta_m:B_m \to A$ the inclusion maps, and $\psi_m:B_m \simeq B_m \otimes \cD$ (these exist since $\cD \into (B_1)_\omega \cap A'$ implies that $\cD \into (B_m)_\omega \cap B_m'$), our unitaries satisfy the hypothesis of Proposition \ref{countablerelativeintertwining} and therefore $\phi$ is approximately unitarily equivalent to a *-isomorphism $\Phi: A \simeq A \otimes \cD$ such that $\Phi(B_m) = B_m \otimes \cD$ for all $m$. + \end{proof} + \end{theorem} + + + The above works since norm ultrapowers have the property that unitaries lift to sequences of unitaries.\footnote{If $u = (u_n) \in A_\omega$, then $\{n \in \bN \mid \|u_n^*u_n - 1\|\|u_nu_n^* - 1\| < 1\} \in \omega$. If $n$ is in the set, replace $u_n$ with the unitary part of its polar decomposition, and replace $u_n$ with 1 otherwise.} Tracial ultrapowers of $\II_1$ von Neumann algebras also have this property.\footnote{The tracial ultrapower of a $\II_1$ von Neumann algebra is again a $\II_1$ von Neumann algebra. Therefore if $u \in \cM^\omega$ is unitary, it is of the form $e^{ia}$ for some $a = a^* \in \cM^\omega$. Lift $a$ to a sequence $(a_n)$ of self-adjoints in $\cM$ and note that $u = (e^{ia_n})$, so that $u$ has a unitary lift.} Consequently if we work with the 2-norm $\|x\|_2 = \tau(x^*x)$ where $\tau$ is the unique trace on a $\II_1$ factor, all of the above arguments with the C*-norm replaced by $\|\cdot\|_2$ will yield back Bisch's result \cite[Theorem 3.1]{Bisch90}, provided we have the appropriate separability conditions. + + \begin{theorem} + Let $\cN \subseteq \cM$ be an inclusion of $\II_1$ factors with separable preduals. Then $\cR \into \cN^\omega \cap \cM'$ if and only if there exists an isomorphism $\Phi: \cM \to \cM \ov{\otimes} \cR$ such that $\Phi(\cN) = \cN \ov{\otimes} \cR$. + \end{theorem} + + \subsection{Existence of $\cD$-stable embeddings}\label{ssection:dse-existence} + + We move to discuss the existence of $\cD$-stable embeddings. First we show that each unital embedding of unital, separable $\cD$-stable C*-algebras is approximately unitarily equivalent to a $\cD$-stable embedding. From this it will follow that there are many $\cD$-stable embeddings. + + \begin{lemma}\label{adudstable} + Let $\cD$ be strongly self-absorbing. If $\iota: B \into A$ is a unital, $\cD$-stable inclusion of separable C*-algebras and $u \in U(A)$, then $\Ad_u \circ \iota: B \into A$ is $\cD$-stable. + \begin{proof} + Apply Lemma \ref{aduaapproximateembeddings}. + \end{proof} + \end{lemma} + + \begin{prop}\label{allembeddingsareue} + Let $\cD$ be strongly self-absorbing, $A,B$ be unital separable $\cD$-stable C*-algebras and let $\iota: B \into A$ be an embedding. Then $\iota$ is approximately unitarily equivalent to a $\cD$-stable embedding $B \into A$. + \begin{proof} + As $A,B$ are $\cD$-stable, there are isomorphisms +\begin{equation} + \phi: B \simeq B \otimes \cD \text{ and }\psi: A \simeq A \otimes \cD +\end{equation} + which are approximately unitarily equivalent to the first factor embeddings $b \mapsto b \otimes 1_\cD, b \in B$ and $a \mapsto a \otimes 1_\cD, a \in A$ respectively. As $\iota \otimes \id_\cD: B \otimes \cD \into A \otimes \cD$ is $\cD$-stable by Lemma \ref{trivialinclusion}, +\begin{equation} + \sigma := \psi^{-1} \circ (\iota \otimes \id_\cD) \circ \phi: B \into A +\end{equation} + is $\cD$-stable by Lemma \ref{conjugations}. Now we show that $\sigma$ is approximately unitarily equivalent to $\iota$. Let $\cF \subseteq B$ be finite and $\ee > 0$. Let $u \in U(B \otimes \cD)$ be such that $u^*(b \otimes 1_\cD)u \approx_\frac{\ee}{2} \phi(b)$ for $b \in \cF$ and $v \in U(A \otimes \cD)$ be such that $v^*(\iota(b) \otimes 1_\cD)v \approx_{\frac{\ee}{2}} \psi \circ \iota (b)$ for $b \in \cF$. Set $w = \psi^{-1}(\iota \otimes \id_\cD(u))^*\psi^{-1}(v) \in U(A)$. Then for $b \in \cF$, +\begin{equation} +\begin{split} + w^*\sigma(b)w &= \psi^{-1}(v)^*\psi^{-1}(\iota \otimes \id_\cD(u\phi(b)u^*))\psi^{-1}(v) \\ + &\approx_{\frac{\ee}{2}} \psi^{-1}(v)^*\psi^{-1}(\iota \otimes \id_\cD(b \otimes 1_\cD))\psi^{-1}(v) \\ + &= \psi^{-1}(v)^*\psi^{-1}(\iota(b) \otimes 1_\cD)\psi^{-1}(v) \\ + &\approx_{\frac{\ee}{2}} \psi^{-1}(\psi(\iota(b))) \\ + &= \iota(b). +\end{split} +\end{equation} + + \end{proof} + \end{prop} + + + \begin{cor}\label{pointnormdensity} + Let $\cD$ be strongly self-absorbing. The set of $\cD$-stable embeddings $B \into A$ of unital, separable, $\cD$-stable C*-algebras is point-norm dense in the set of embeddings $B \into A$. + \begin{proof} + Every embedding is approximately unitarily equivalent to a $\cD$-stable embedding. As $\cD$-stability of an embedding is preserved if one composes with $\Ad_u$, it follows that every embedding is the point-norm limit of $\cD$-stable embeddings. + \end{proof} + \end{cor} + + \begin{remark}\label{homomorphisminsteadofembedding} + We note that it is not actually necessary that $\iota$ is an embedding. If $\pi: B \to A$ is any unital *-homomorphism between unital, separable, $\cD$-stable C*-algebras, then $\pi$ is approximately unitarily equivalent to a *-homomorphism $\pi': B \to A$ such that $\pi'(B) \subseteq A$ is $\cD$-stable. Consequently the set of unital *-homomorphisms $\pi: B \to A$ with $\pi(B) \subseteq A$ being $\cD$-stable is in fact dense in the set of unital *-homomorphisms $B \to A$. + \end{remark} + + Later on, there will be some examples of non-$\cD$-stable embeddings between $\cD$-stable C*-algebras. Consequently, despite the fact $\cD$-stable embeddings are point-norm dense, the set of $\cD$-stable embeddings need not coincide with the set of all embeddings $B \into A$. Another clear consequence is that despite $\cD$-stability of an embedding being closed under conjugation by a unitary, it is not true that it is preserved under approximate unitary equivalence (in fact, the examples in question show that $\cD$-stability is not even preserved under asymptotic unitary equivalence). We finish with a corollary about embeddings into the Cuntz algebra $\cO_2$ \cite{Cuntz77}. + + \begin{cor} + Let $B$ be a unital, separable, exact $\cD$-stable C*-algebra, where $\cD$ is strongly self-absorbing. Then there is a $\cD$-stable embedding $B \into \cO_2$. + \begin{proof} + As $\cD$ is unital, simple, separable and nuclear by \cite[Section 1.6]{TomsWinter07}, $\cO_2 \simeq \cO_2 \otimes \cD$ and $B \into \cO_2$ unitally by Theorem 3.7 and Theorem 2.8 of \cite{KirchbergPhillips00} respectively. The above results then yield a $\cD$-stable embedding $B \into \cO_2$. + \end{proof} + \end{cor} + + We include this last result about the classification of morphisms via functors. + + \begin{theorem} + Let $\cD$ be strongly self-absorbing and let $F$ be a functor from a class of unital, separable, $\cD$-stable C*-algebras satisfying the following. + \begin{enumerate} + \item[(E)] If there exists a morphism $\Phi: F(B) \to F(A)$, then there exists a *-homomorphism $\phi: B \to A$ such that $F(\phi) = \Phi$. + \item[(U)] If $\phi,\psi: B \to A$ are *-homomorphisms which are approximately unitarily equivalent, then +\begin{equation} + F(\phi) = F(\psi). +\end{equation} + \end{enumerate} + Then whenever there is a morphism $\Phi: F(B) \to F(A)$, there exists $\phi: B \to A$ such that $F(\phi) = \Phi$ and $\phi(B) \subseteq A$ is $\cD$-stable. + Moreover, $\phi$ is unique up to approximate unitary equivalence. + \begin{proof} + By the existence (E), there exists a *-homomorphism $\phi: B \to A$. Now by Proposition \ref{allembeddingsareue} (Remark \ref{homomorphisminsteadofembedding} allows us to work with general *-homomorphisms), there exists a *-homomorphism $\phi': B \to A$ which is approx\hyp{}imately unitarily equivalent to $\phi$ and $\phi'(B) \subseteq A$ is $\cD$-stable. + Uniqueness (U) gives that this is unique up to approximate unitary equivalence. + \end{proof} + \end{theorem} + + \subsection{Permanence properties}\label{ssection:dse-permanence} + We now discuss some permanence properties. + + + \begin{lemma} +Let $\cD$ be strongly self-absorbing. +Suppose that $\iota_i: B_i \into A_i, i=1,2$ are $\cD$-stable inclusions. Then $\iota_1 \oplus \iota_2: B_1 \oplus B_2 \into A_1 \oplus A_2$ is $\cD$-stable. + \begin{proof} + Let $\Phi_i: A_i \simeq A_i \otimes \cD$ be isomorphisms such that $\Phi_i\circ \iota_i(B_i) = \iota_i(B_i) \otimes \cD$ and consider +\begin{equation} + \Phi: A_1 \oplus A_2 \simeq (A_1 \oplus A_2) \otimes \cD +\end{equation} + given by the composition +\begin{equation} + \begin{tikzcd} +A_1 \oplus A_2 \arrow[r, "\Phi_1 \oplus \Phi_2"] & (A_1 \otimes \cD) \oplus (A_2 \otimes \cD) \arrow[r, "\simeq"] & (A_1 \oplus A_2) \otimes \cD +\end{tikzcd} +\end{equation} +where the last isomorphism follows from (finite) distributivity of the min-tensor. Then we see that +\begin{equation} +\Phi(\iota_1(B_1) \oplus \iota_2(B_2)) = \left(\iota_1(B_1) \oplus \iota_2(B_2)\right) \otimes \cD. +\end{equation} + \end{proof} + \end{lemma} + + \begin{lemma} +Let $\cD$ be strongly self-absorbing. + Suppose that $\iota_i: B_i \into A_i, i=1,2$ are inclusions and that at least one of $\iota_1$ or $\iota_2$ is $\cD$-stable. Then $\iota_1 \otimes \iota_2: B_1 \otimes B_2 \into A_1 \otimes A_2$ is $\cD$-stable. + \begin{proof} + We prove this if $\iota_2$ is $\cD$-stable, and a symmetric argument will yield the result if $\iota_1$ is. Let $\Phi_2: A_2 \simeq A_2 \otimes \cD$ be such that $\Phi_2 \circ \iota_2(B_2) = \iota(B_2) \otimes \cD$. Taking +\begin{equation} + \Phi := \id_{A_1} \otimes \Phi_2: A_1 \otimes A_2 \simeq A_1 \otimes A_2 \otimes \cD, +\end{equation} + we have that +\begin{equation} + \Phi(\iota_1(B_1) \otimes \iota_2(B_2)) = \iota_1(B_1) \otimes \iota_2(B_2) \otimes \cD. +\end{equation} + \end{proof} + \end{lemma} + + \begin{prop} + Let $\cD$ be strongly self-absorbing. +Suppose that we have increasing sequences of unital separable C*-algebras $(B_n)$ and $(A_n)$ such that $B_n \subseteq A_n$ unitally. + \begin{comment} +\begin{equation} + \begin{tikzcd} +B_1 \arrow[d, hook] \arrow[r, hook] & B_2 \arrow[d, hook] \arrow[r] & \cdots \arrow[r] & B \arrow[d, hook] \\ +A_1 \arrow[r, hook] & A_2 \arrow[r] & \cdots \arrow[r] & A , +\end{tikzcd} +\end{equation} +\end{comment} +Let $B = \ov{\cup_nB_n}$ and $A = \ov{\cup_n A_n}$. If $B_n \subseteq A_n$ is $\cD$-stable for all $n$, then $B \subseteq A$ is $\cD$-stable. + \begin{proof} + This follows from Proposition \ref{approximateembeddinginductivelimit}, together with Lemma \ref{lem:local-equiv-global} and Theorem \ref{equivDstable}. + \end{proof} + \end{prop} + + + Lastly we'll discuss unital inclusions $B \subseteq A$ of $C(X)$ algebras, where $X$ is a compact Hausdorff space. We show that if $X$ has finite covering dimension, then such an inclusion is $\cD$-stable if and only if the inclusion $B_x \subseteq A_x$ along each fibres is $\cD$-stable. + + \begin{lemma}\label{imageofinclusion} + Let $\cD$ be strongly self-absorbing. +Suppose that $B_i \subseteq A_i$ are unital inclusions, for $i=1,2$, and $\psi: A_1 \to A_2$ is a surjective *-homomorphism such that $\psi(B_1) = B_2$. If $B_1 \subseteq A_1$ is $\cD$-stable, then so is $B_2 \subseteq A_2$. + \begin{proof} + We note that $\psi$ induces a *-homomorphism +\begin{equation} + \tilde{\psi}: (B_1)_\omega \cap A_1' \to (B_2)_\omega \cap A_2' +\end{equation} + and consequently if $\xi: \cD \into (B_1)_\omega \cap A_1'$, we have a unital *-homomorphism +\begin{equation} + \eta := \tilde{\psi} \circ \xi: \cD \to (B_2)_\omega \cap A_2'. +\end{equation} + $\eta$ is automatically injective since $\cD$ is simple. + \end{proof} + \end{lemma} + + +Rephrasing the above in terms of commutative diagrams, it says that if we have a commutative diagram +\begin{equation} +\begin{tikzcd} +A_1 \arrow[r, two heads] & A_2 \\ +B_1 \arrow[r, two heads] \arrow[u, hook] & B_2 \arrow[u, hook] +\end{tikzcd} +\end{equation} +where the left inclusion is $\cD$-stable, then the right inclusion is $\cD$-stable as well. + +Now we consider many of the results discussed in \cite[Section 4]{HirshbergRordamWinter07}, except for inclusions of C*-algebras. + +\begin{defn} + Let $X$ be a compact Hausdorff space. A $C(X)$-algebra is a C*-algebra $A$ endowed with a unital *-homomorphism $C(X) \to \cZ(\cM(A))$, where $\cZ(\cM(A)$ is the center of the multiplier algebra $\cM(A)$ of $A$. + \end{defn} + + If $Y \subseteq X$ is a closed subset, we set $I_Y := C_0(X \sm Y)A$, which is a closed two-sided ideal in $A$. We denote $A_Y := A/A_Y$ and the quotient map $A \to A_Y$ by $\pi_Y$. For an element $a \in A$, we write $a_Y := \pi_Y(a)$ and if $Y$ consists of a single point $x$, we write $A_x,I_x,\pi_x$ and $a_x$. We say that $A_x$ is the fibre of $A$ at $x$. We note that $A_X = A$. + + If $B \subseteq A$ is a unital inclusion and $\theta:_A: C(X) \to A, \theta_B: C(X) \to B$ are morphisms witness $A$ and $B$ as $C(X)$-algebras, respectively, we say that $B \subseteq A$ is an inclusion of $C(X)$-algebras if + \begin{equation} + \begin{tikzcd} +B \arrow[r, hook] & A \\ +C(X) \arrow[u, "\theta_B"] \arrow[ru, "\theta_A"'] & +\end{tikzcd} + \end{equation} + commutes. Note that $\theta_B(B) \subseteq \cZ(A)$. We note that when discussion an inclusion of fibres $B_Y \subseteq A_Y$ we are considering $B_Y := \pi^A_Y(B) \subseteq \pi^A_Y(A) =: A_Y$, where $\pi_Y^A: A \to A_Y$ is the associated quotient map. + + + \begin{remark}[Upper semi-continuity]\label{rem:up-semi-cts} + In \cite[Section 1.3]{HirshbergRordamWinter07}, it was pointed out that the norm on a $C(X)$-algebra $A$ is upper semi-continuous. This meaning that, fixing some $a \in A$, the function $x \mapsto \|a_x\|$ from $X$ to $\bR$ is upper semi-continuous (as it is the infimum of a family of continuous functions), and consequently the set $\{x \in X \mid \|a_x\| < \ee\} \subseteq X$ is open for all $a \in A$ and $\ee > 0$. + \end{remark} + + + \begin{comment} + For two unital $C(X)$-algebras $A,B$, let $\theta_A: C(X) \to Z(A)$ and $\theta_B: C(X) \to Z(B)$ be the unital *-homomorphisms in the above definition. We say that a *-homomorphism $\phi: B \to A$ is a $C(X)$-morphism if +\begin{equation} + \phi \circ \theta_B = \theta_A. +\end{equation} + Note that for a closed subset $Y \subseteq X$, a $C(X)$-morphism induces a *-homomorphism $\phi_Y: B_Y \to A_Y$ given by +\begin{equation} + \phi_Y(\pi_Y(b)) = \pi_Y(\phi (b)). +\end{equation} + This gives a morphism of short exact sequences: +\begin{equation} + \begin{tikzcd} +0 \arrow[r] & I_Y^{B} \arrow[r] \arrow[d, "\phi|_{I_Y^{B}}"] & B \arrow[r, "\pi_Y^{B}"] \arrow[d, "\phi"] & B_Y \arrow[r] \arrow[d, "\phi_Y"] & 0 \\ +0 \arrow[r] & I_Y^{A} \arrow[r] & A \arrow[r, "\pi_Y^A"'] & A_Y \arrow[r] & 0. +\end{tikzcd} +\end{equation} +%\begin{comment} +Given an embedding $\phi: B \to A$ of $C(X)$-algebras, we will be interested in the fibrewise inclusions $\phi_Y(B_Y) \subseteq A_Y$. +\end{comment} + +We note that Lemma \ref{imageofinclusion} gives that if $B\subseteq A$ is $\cD$-stable and $Y \subseteq X$ is closed, then $B_Y \subseteq A_Y$ is automatically $\cD$-stable as well since we have the commuting diagram +\begin{equation} +\begin{tikzcd} +A \arrow[r, "\pi_Y"] & A_Y \\ +B \arrow[u, hook] \arrow[r, "\pi_Y|_B"'] & B_Y. \arrow[u, hook] +\end{tikzcd} +\end{equation} +The converse needs a bit of work. This is the embedding-analogue of the beginning of \cite[Section 4]{HirshbergRordamWinter07}. We discuss how the proofs can be adapted and often omit approximations that were otherwise done there. We want a version of \cite[Lemma 4.5]{HirshbergRordamWinter07}, which is a result about \emph{gluing} c.c.p. maps together along fibres. In our setting, we are only interested in u.c.p. maps, and we want to show that if we \emph{glue} two u.c.p. maps together whose images are contained in some $C(X)$-subaglebra $B$, then the \emph{glued} map also has image contained in $B$. We borrow their Definition 4.2. + +\begin{defn} +Let $A$ be a unital $C(X)$-algebra, for a compact Hausdorff space $X$, and let $D$ be a unital C*-algebra. Let $\phi: D \to A$ is a u.c.p. map and $Y \subseteq X$ a closed subset. If $\cF \subseteq D, \cG \subseteq A$ are finite and $\ee > 0$, we say that $\phi$ is $(\cF,\ee,\cG)$-good for $Y$ if +\begin{enumerate} +\item $([\phi(d),a])_Y \approx_\ee 0$ and +\item $\phi(dd')_Y \approx_\ee \phi(d)_Y\phi(d')_Y$ +\end{enumerate} +whenever $d,d' \in \cF$ and $a \in \cG$. If $X = [0,1]$, $Y \subseteq X$ is a closed interval, $\cF' \supseteq \cF$ is another finite set and $0 < \ee' < \ee$, we say that $\phi$ is $(\cF,\ee,\cG;\cF',\ee')$-good for $Y$ if $\phi$ is $(\cF,\ee,\cG)$-good for $Y$ and there exists some closed neighbourhood $V$ of the endpoints of $Y$ such that $\phi$ is $(\cF',\ee',\cG)$-good for $V$. +\end{defn} + +First we need a lemma that follows as a consequnce of $\cD$-stability. It is the embedding analogue of \cite[Proposition 4.1]{HirshbergRordamWinter07}. + +\begin{lemma}\label{lem:kappa-mu} +Let $\cD$ be strongly self-absorbing, and $B \subseteq A$ be a unital, $\cD$-stable inclusion of separable C*-algebras. Then for any $\cG \subseteq A$ finite and $\ee > 0$, there exist unital *-homomorphisms $\kappa: A \to A$ and $\mu: \cD \to B$ such that +\begin{enumerate} +\item $\kappa(B) \subseteq B$, +\item $[\kappa(A),\mu(\cD)] = 0$, +\item $\kappa(a) \approx_\ee a$ for all $a \in \cG$. +\end{enumerate} +\begin{proof} +The proof is essentially the same as the proof of (a) $\Rightarrow (c)$ in \cite[Proposition 4.1]{HirshbergRordamWinter07}. As $B \subseteq A$ is $\cD$-stable, let us identify $B \subseteq A$ with $B \otimes \cD \subseteq A \otimes \cD$. As $\cD$ is strongly self-absorbing, \cite[Theorem 2.3]{TomsWinter07} gives a sequence $(\phi_n)$ of *-homomorphisms $\phi_n: \cD \otimes \cD \to \cD$ such that +\begin{equation} +\phi_n(d \otimes 1_\cD) \to d \text{ for all } d \in \cD. +\end{equation} +Define $\kappa_n: A \otimes \cD \to A \otimes \cD$ by +\begin{equation} +\kappa_n:= (\id_A \otimes \phi) \circ (\id_A \otimes \id_\cD \otimes 1_\cD), +\end{equation} +and $\mu_n: \cD \to B \otimes \cD$ by +\begin{equation} +\mu_n := (\id_B \otimes \phi_n) \circ (1_A \otimes 1_\cD \otimes \id_\cD). +\end{equation} +Then taking $n$ large enough and letting $\kappa$ and $\mu$ be $\kappa_n$ and $\mu_n$ respectively, its clear that $\kappa(B \otimes \cD) \subseteq B \otimes \cD$, $[\kappa(A),\mu(\cD)] = 0$ and that $\kappa(a) \approx_\ee a$ whenever $a$ is in some prescribed finite subset $\cG \subseteq A$ and $\ee > 0$ is some prescribed error. +\end{proof} +\end{lemma} + +\begin{lemma} +Let $\cD$ be strongly self-absorbing and $A$ be a unital, separable $C([0,1])$-algebra. Suppose $\cF \subseteq \cD, \cG \subseteq A$ are finite self-adjoint subsets of contractions with $1_\cD \in \cF$. Suppose that we have points $0 \leq r < s < t \leq 1$ and two u.c.p. maps $\rho,\sigma: \cD \to A$ which are $(\cF,\ee,\cG)$-good for $[r,s],[s,t]$ respectively. Suppose that $A_s$ is $\cD$-stable. + +Then there are u.c.p. maps $\rho',\sigma': \cD \to A$ which are $(\cF,\ee,\cG)$-good for $[r,s],[s,t]$ respectively, and u.c.p. maps $\nu_{\rho'},\nu_{\sigma'}: \cD \to A,\mu_{\rho'},\mu_{\sigma'}: \cD \otimes \cD \to A$ such that $\nu_{\rho'},\nu_{\sigma'}$ are $(\cF,3\ee,\cG)$-good for some interval $I \subseteq (r,t)$ containing $s$ in its interior, and such that for any $a \in \cG, d,d' \in \cF$, we have +\begin{enumerate} +\item $([\rho'(d),\nu_{\rho'}(d')])_I \approx_{2\ee} 0$ +\item $([\sigma'(d),\nu_{\sigma'}(d')])_I \approx_{2\ee} 0$ +\item $\rho'(d)_I\nu_{\rho'}(d')_I \approx_\ee \mu_{\rho'}(d \otimes d')_I$ +\item $\sigma'(d)_I\nu_{\sigma'}(d')_I \approx_\ee \mu_{\sigma'}(d \otimes d')_I$ +\item $\nu_{\rho'}(d)_I \approx_{2\ee} \nu_{\sigma'}(d)_I$. +\end{enumerate} +If $\rho,\sigma$ are $(\cF,\ee,\cG;\cF',\ee)$-good for $[r,s],[s,t]$ respectively, for some finite $\cF' \supseteq \cF$ set of contractions and for some $0 < \ee' < \ee$, then we can arrange so that $\rho',\sigma',\nu_{\rho'},\nu_{\sigma'}$ are $(\cF',3\ee',\cG)$-good for the interval $I$, and that the above five conditions hold with $\ee'$ in place of $\ee$ and $\cF'$ in place of $\cF$. + +Moreover, if $B \subseteq A$ is a unital inclusion of $C([0,1])$-algebras such that $\rho(\cD) \subseteq B, \sigma(\cD) \subseteq B$ and $B_s \subseteq A_s$ is $\cD$-stable, then the images of all $\rho',\sigma',\mu_{\rho'},\mu_{\sigma'}$ are contained in $B$ (as are the images of $\nu_{\rho'}$ and $\nu_{\sigma'}$). +\begin{proof} +This is \cite[Lemma 4.4]{HirshbergRordamWinter07}, except we've replaced c.c.p. maps with u.c.p. maps. One can easily check that the resulting maps are u.c.p. maps. + +As for the ``moreover'' part, which is the only addition besides the unitality, we outline the definitions of these maps to show that the images of $\rho',\sigma',\mu_{\rho'},\mu_{\sigma'}$ are contained in $B$. As $B_s \subseteq A_s$ is $\cD$-stable, we can find $\kappa: A_s \to A_s$ and $\mu: \cD \to B_s$ as in Lemma \ref{lem:kappa-mu}, where $\kappa(a_s) \approx a_s$ for an appropriate error whenever $a \in \cG$. We use Choi-Effros to find u.c.p. lifts $\tilde{\rho},\tilde{\sigma}: \cD \to B$ for the maps $\kappa \circ \pi_s \circ \rho$ and $\kappa \circ \pi_s \circ \sigma$ respectively (note that $\kappa \circ \pi_s \circ \rho$ and $\kappa \circ \pi_s \circ \sigma$ lie in $B_s$, which is a *-homomorphism image of $B$). One then defines piece-wise linear functions $f,g: [0,1] \to [0,1]$ which attain both values 0 and 1 at the end points (their definition is not important to show the ``moreover'' part). Then $\rho',\sigma'$ are defined as +\begin{equation} +\rho'(d) := (1-f)\cdot \rho(d) + f\cdot\tilde{\rho}(d) \text{ and } \sigma'(d) := (1-g)\cdot\sigma(d) + g\cdot \tilde{\sigma}(d) +\end{equation} +Clearly $\rho',\sigma'$ take values in $B$ as $\rho,\tilde{\rho},\sigma,\tilde{\sigma}$ all do and $(1-f),f,(1-g),g$ are in $B$. Now we define u.c.p. maps $\tilde{\mu}_{\rho'},\tilde{\mu}_{\sigma'}: \cD \otimes \cD \to B_s$ by +\begin{equation} +\tilde{\mu}_{\rho'}(d \otimes d') := \rho'(d)_s\mu(d') \text{ and }\tilde{\mu}_{\sigma'}(d \otimes d') := \sigma'(d)_s\mu(d'). +\end{equation} +Now by Choi-Effros, we can take u.c.p. lifts $\mu_{\rho'}$ and $\mu_{\sigma'}$ of $\tilde{\mu}_{\rho'}$ and $\mu_{\sigma'}$, respectively. As the images of $\tilde{\mu}_{\rho'}$ and $\mu_{\sigma'}$ lie in $B_s$, the images of $\mu_{\rho'}$ and $\mu_{\sigma'}$ will lie in $B$. +\end{proof} +\end{lemma} + +\begin{lemma}\label{lem:exist-F-ee} +Let $A$ be a unital, separable $C([0,1])$-algebra. Suppose $\cF \subseteq \cD, \cG \subseteq A$ are finite self-adjoint subsets with $1_\cD \in \cF$ and $\ee > 0$. There exists $0 < \ee' < \ee$ and a finite subset $\cF' \supseteq \cF$ such that if $\rho,\sigma: \cD \to A$ are u.p.c. maps and $0 \leq r < s < t \leq 1$ are points such that $\rho$ is $(\cF,\ee,\cG;\cF',\ee')$-good for $[r,s]$, $\sigma$ is $(\cF,\ee,\cG;\cF',\ee')$-good for $[s,t]$ and $A_s$ is $\cD$-stable, then there is a u.c.p. map $\psi: \cD \to A$ which is $(\cF,\ee,\cG;\cF',\ee')$-good for $[r,t]$. + +Moreover, if $B \subseteq A$ is a unital inclusion of $C([0,1])$-algebras such that $\rho(\cD) \subseteq B$, $\sigma(\cD) \subseteq B$ and $B_s \subseteq A_s$ is $\cD$-stable, then $\psi(\cD) \subseteq B$. +\begin{proof} +The first part is \cite[Lemma 4.5]{HirshbergRordamWinter07}, except we've replaced c.c.p. maps with u.c.p. maps. One has to check that the resulting $\psi$ is unital, but this follows easily if $\rho$ and $\sigma$ are. + +We outline the construction of $\psi$ to show unitality, as it will also be useful to show the ``moreover'' part, which is the only real addition. Let $u \in C([0,1],\cD \otimes \cD)$ be a path of unitaries such that $u_0 = 1_{\cD \otimes \cD}$ and +\begin{equation} +u_1(d \otimes 1_\cD)u_1^* \approx_{\frac{\ee}{4}} 1_\cD \otimes d. +\end{equation} +We replace $\rho,\sigma$ with $\rho',\sigma'$ as in the above lemma and this yields u.c.p. maps $\mu_\rho,\mu_\sigma$ satisfying the hypotheses above for some interval $I \subseteq (s,t)$ with $s$ in its interior. Define +\begin{equation} +\phi_\rho,\phi_\sigma: C([0,1]) \otimes \cD \otimes \cD \to A +\end{equation} +by +\begin{equation} +\begin{split} +\phi_\rho(f \otimes d \otimes d') &:= f\cdot \mu_\rho(d \otimes d') \\ +\phi_\sigma(f \otimes d \otimes d') &:= f\cdot \mu_\sigma(d \otimes d'). +\end{split} +\end{equation} +Note that these maps are unital. Take piece-wise linear functions $h_1,h_2,h_3,h_4: [0,1] \to [0,1]$ which sum to 1 (their specific form does not matter to show unitality of $\psi$ nor the ``moreover'' part) and $g_\rho,g_\sigma: [0,1] \to [0,1]$ which sum to 1 (again, their specific form does not matter to show unitality of $\psi$ nor the ``moreover'' part). Define unitaries $u_\rho,u_\sigma \in C([0,1]) \otimes \cD \otimes \cD \simeq C([0,1],\cD \otimes \cD)$ by +\begin{equation} +u_{\rho x} := u_{g_\rho(x)} \text{ and } u_{\sigma x} := u_{g_\sigma(x)}. +\end{equation} +Now define $\zeta_\rho,\zeta_\sigma: \cD \to A$ by +\begin{equation} +\begin{split} +\zeta_\rho(d) &:= \phi_\rho(u_\rho(1_{C([0,1])} \otimes d \otimes 1_\cD)u_\rho^*) \\ +\zeta_\sigma(d) &:= \phi_\sigma(u_\sigma(1_{C([0,1])} \otimes d \otimes 1_\cD)u_\sigma^*), +\end{split} +\end{equation} +which are clearly unital. Finally the map $\psi: \cD \to A$ is defined by +\begin{equation} +\psi(d) := h_1\cdot\rho(d) + h_2\cdot \zeta_\rho(d) + h_3\cdot \zeta_\sigma(d) + h_4\cdot \sigma(d). +\end{equation} +Clearly $\psi$ is unital. + +Now for the ``moreover'' part. If $\rho(\cD) \subseteq B$ and $\sigma(\cD) \subseteq B$, clearly the first and fourth terms in the definition of $\psi$ will lie in $B$. So it suffices to show that $\zeta_\rho(\cD) \subseteq B$ and $\zeta_\sigma(\cD) \subseteq B$, and for this it suffices to show that $\mu_\rho(\cD \otimes \cD) \subseteq B$ and $\mu_\sigma(\cD \otimes \cD) \subseteq B$ (since $h_1,h_2,h_3,h_4$ all lie in $B$). But this follows from the ``moreover'' part of the previous lemma. +\end{proof} +\end{lemma} + + With this, we get the analogue of their Theorem 4.6, the proof being essen\hyp{}tially the same as well, except we insist that the our u.c.p. maps commute with a prescribed finite subset of $A$. + +\begin{prop} +Let $\cD$ be strongly self-absorbing, and $X$ be a compact Hausdorff space with finite covering dimension. Suppose that $B \subseteq A$ is a unital inclusion of $C(X)$-algebras. Then $B_x \subseteq A_x$ is $\cD$-stable for all $x \in X$ if and only if $B \subseteq A$ is $\cD$-stable. + \begin{proof} + As previously mentioned, if $B \subseteq A$ is $\cD$-stable, then $B_x \subseteq A_x$ is $\cD$-stable for all $x$. + + For the converse, the proof is essentially the same as \cite[Theorem 4.6]{HirshbergRordamWinter07}. Using the arguments there, one can simplify to the case where we can argue this for $C([0,1])$-algebras (by using \cite[Theorem V.3]{HurewiczWallman}, which says that a compact space of dimension $\leq n$ is homeomorphic to a subset of $[0,1]^{2n+1}$, and then working component-wise). Now for $\cF \subseteq \cD, \cG \subseteq A$ and $\ee > 0$, let $\cG_x := \{a_x \mid a \in \cG\}$. Without loss of generality suppose that $\cF^*=\cF,\cG^*=\cG$ and that $1_\cD \in \cF$. Let $\cF',\ee'$ be as in Lemma \ref{lem:exist-F-ee}. + + By $\cD$-stability of the inclusion $B_x \subseteq A_x$ there are u.c.p. $(\cF',\ee',\cG_x)$-embed\hyp{}dings $\psi_x: \cD \to B_x \subseteq A_x$ which lift by Choi-Effros to u.c.p. maps $\psi_x': \cD \to B$. The norm is upper semi-continuous (Remark \ref{rem:up-semi-cts}), and this yields intervals $I_x \subseteq [0,1]$ such that $\psi_x'$ is $(\cF',\ee',\cG)$-good for $\ov{I_x}$. Note that $\psi_x'$ being $(\cF',\ee',\cG)$-good for the whole of $I_x$ implies that it is $(\cF,\ee,\cG;\cF',\ee')$-good for $\ov{I_x}$. + Compactness then allows us to split the interval as +\begin{equation} + 0 = t_0 < t_1 < \dots < t_n = 1 +\end{equation} + and to take $\psi_i: \cD \to B$ u.c.p. which are $(\cF,\ee,\cG;\cF',\ee')$-good for $[t_{i-1},t_i]$ for $i=1,\dots,n$ ($\psi_i = \psi_x'$ for some $x \in [0,1]$). Now by repeatedly using the gluing lemma (Lemma \ref{lem:exist-F-ee}) to glue these maps together, we can find a u.c.p. map $\psi: \cD \to B$ which is an $(\cF,\ee,\cG)$-embedding. + \end{proof} +\end{prop} + + + \section{Crossed products}\label{section:dse-crossed-products} + + In this section we discuss how inclusions coming from non-commutative dynamics fit into the framework of a tensorially absorbing inclusions. We'll shortly discuss group actions $G \act^\alpha A$ with Rokhlin properties and consider the inclusion of a C*-algebra in its crossed product $A \subseteq A \rtimes_\alpha G$, as well as the inclusion of the fixed point subalgebra of the action in the C*-algebra $A^\alpha \subseteq A$. We then discuss diagonal inclusions associated to certain group actions. + + This first result says that if we have an isomorphism $A \simeq A \otimes \cD$ which is $G$-equivariant with respect to an action point-wise fixing the right tensor factor, then the corresponding inclusion $A \subseteq A \rtimes_{r,\alpha} G$ is $\cD$-stable. + + \begin{prop} + Let $G \act^\alpha A$ be an action of a countable discrete group on a unital separable C*-algebra. Suppose that $\alpha \simeq \alpha \otimes \id_\cD$, that is, there is an isomorphism $\Phi: A \simeq A \otimes \cD$ such that +\begin{equation} + \begin{tikzcd} +A \arrow[d, "\alpha_g"'] \arrow[r, "\Phi"] & A \otimes \cD \arrow[d, "\alpha_g \otimes \id_\cD"] \\ +A \arrow[r, "\Phi"'] & A \otimes \cD +\end{tikzcd} +\end{equation} +commutes for all $g \in G$. Then $A \subseteq A \rtimes_{r,\alpha} G$ is $\cD$-stable. + \begin{proof} + Let $\psi: \cD \simeq \cD^{\otimes \infty}$ and let $\phi_n: \cD \to \cD^{\otimes \infty}$ be the $n$th factor embedding: +\begin{equation} + \phi_n(d) := 1_\cD^{\otimes n-1} \otimes d \otimes 1_\cD^{\otimes \infty}. +\end{equation} + We claim that $\xi(d) := (\Phi^{-1}(1_A \otimes \psi^{-1} \circ \phi_n(d)))_n: \cD \to A_\omega$ is an embedding such that $\xi(\cD) \subseteq A_\omega \cap A'$ and $(\alpha_g)_\omega \circ \xi = \xi$ for all $g \in G$ -- that is, $\xi$ is an embedding $\cD \into A_\omega \cap (A \rtimes_{r,\alpha} G)'$. The first claim is obvious, so we prove the second. We have +\begin{equation} +\begin{split} + \|\alpha_g(\Phi^{-1}(1_A &\otimes \psi^{-1}(\phi_n(d)))) - \Phi^{-1}(1_A \otimes \psi^{-1}(\phi_n(d)))\| \\ + &= \|\Phi \circ \alpha_g(\Phi^{-1}(1_A \otimes \psi^{-1}(\phi_n(d)))) - \Phi(\Phi^{-1}(1_A \otimes \psi^{-1}(\phi_n(d))))\| \\ + &= \|\alpha_g \otimes \id_\cD(1_A \otimes \psi^{-1}(\phi_n(d))) - 1_A \otimes \psi^{-1}(\phi_n(d)))\| \\ + &=0. +\end{split} +\end{equation} + \end{proof} + \end{prop} + + The next lemma of note is the following. + + \begin{lemma}\label{lem:fixed-point-in-crossed-product-D-stable} + Suppose that $G \act^\alpha A$ is an action of a finite group on a unital separable C*-algebra $A$ such that $A \subseteq A \rtimes_\alpha G$ is $\cD$-stable. Then $A^\alpha \subseteq A \rtimes_\alpha G$ is $\cD$-stable. In particular, if $A \subseteq A\rtimes_\alpha G$ is $\cD$-stable, then $C \simeq C \otimes \cD$ whenever $A^\alpha \subseteq C \subseteq A \rtimes_\alpha \cD$. + \begin{proof} + For an element $(x_n) \in A_\omega \cap (A \rtimes_\alpha G)'$, an easy averaging argument shows that +\begin{equation} + (x_n) = \left(\frac{1}{|G|}\sum_{g \in G} \alpha_g(x_n)\right) +\end{equation} +in $A_\omega$, and the right is clearly point-wise fixed by $\alpha_g$ for all $g \in G$. So $A_\omega \cap (A \rtimes_\alpha G)'$ is actually equal to $(A^\alpha)_\omega \cap (A \rtimes_\alpha G)'$, and the existence of a unital embedding of $\cD$ in $A_\omega \cap (A \rtimes_\alpha G)'$ is in fact equivalent to the existence of a unital embedding of $\cD$ into $(A^\alpha)_\omega \cap (A \rtimes_\alpha G)'$. The result follows. + \end{proof} + \end{lemma} + + + Using the Galois correspondence of Izumi \cite{Izumi02} yields the following. + + \begin{theorem}\label{Galois} + Let $A$ be a unital, simple, separable C*-algebra and let $G \act^\alpha A$ be an action of a finite group by outer automorphisms. If $A \subseteq A \rtimes_\alpha \cD$ is $\cD$-stable, then there exists an isomorphism $\Phi: A \rtimes_\alpha G \simeq (A \rtimes_\alpha G) \otimes \cD$ such that whenever $C$ is a unital C*-algebra satisfying either + \begin{enumerate} + \item $A^\alpha \subseteq C \subseteq A$ or + \item $A \subseteq C \subseteq A \rtimes_\alpha G$, + \end{enumerate} + we have $\Phi(C) = C \otimes \cD$. + \begin{proof} + Applying \cite[Corollary 6.6]{Izumi02} gives the following two correspondences: + \begin{enumerate} + \item there is a one-to-one correspondence between subgroups of $G$ with intermediate C*-algebras $A^\alpha \subseteq C \subseteq A$ given by +\begin{equation} + H \leftrightarrow A^{\alpha_H}; +\end{equation} + \item there is a one-to-one correspondence between subgroups of $G$ and intermediate C*-algebras $A \subseteq C \subseteq A \rtimes_\alpha G$ given by +\begin{equation} + H \leftrightarrow A \rtimes_{\alpha|_H} H. +\end{equation} + \end{enumerate} + In particular, there are only finitely many C*-algebras $C$ between either $A^\alpha \subseteq A$ or $A \subseteq A \rtimes_\alpha G$. As all such lie between the $\cD$-stable inclusion $A^ \alpha \subseteq A \rtimes G$, Theorem \ref{countablymanysubalgebras} yields the desired isomorphism. + \end{proof} + \end{theorem} + + + + \subsection{(Tracial) Rokhlin properties}\label{ssection:dse-tracial-Rokhlin} + + Here we will restrict ourselves to finite groups for simplicity, although many results hold more generally (see \cite{HirshbergWinter07,HirshbergOrovitz13,GardellaHirshberg18}). + + \begin{defn} + Let $A$ be a unital, separable C*-algebra. We say that a finite group action $G \act^\alpha A$ has the Rokhlin property if there are pairwise orthogonal projections $(p_g)_{g \in G} \subseteq A_\omega \cap A'$ summing to $1_{A_\omega}$ such that $(\alpha_g)_\omega(p_h) = p_{gh}$ for $g,h \in G$. + \end{defn} + + \begin{prop} + Let $A$ be a unital, separable $\cD$-stable C*-algebra. If $G \act^\alpha A$ is an action of a finite group with the Rokhlin property, then $A^\alpha \subseteq A \rtimes_\alpha G$ is $\cD$-stable. + \begin{proof} + This follows from \cite[Theorem 3.3]{HirshbergWinter07}, together with Lemma \ref{lem:fixed-point-in-crossed-product-D-stable}. + \end{proof} + \end{prop} + + + \begin{defn} + Let $A$ be a unital, separable C*-algebra. We say that a finite group action $G \act^\alpha A$ has the weak tracial Rokhlin property if for all $\cF \subseteq A$ finite, $\ee > 0$ and $0 \neq a \in A_+$, there are pairwise orthogonal normalized positive contractions $(e_g)_{g \in G} \subseteq A$ such that + \begin{enumerate} + \item $1 - \sum_g e_g \precsim a$;\footnote{For two positive elements $x,y$ in a C*-algebra, we write $x \precsim y$ to mean that $x$ is Cuntz-subequivalent to $y$. That is, there are $(r_n)$ in the C*-algebra such that $r_n^*yr_n \to x$. See \cite[Section 2]{HirshbergOrovitz13}.} + \item $[e_g,x] \approx_\ee 0$ for all $x \in \cF, g \in G$; + \item $\alpha_g(e_h) \approx_\ee e_{gh}$ for all $g,h \in G$. + \end{enumerate} + \end{defn} + + We note that both Rokhlin and weak tracial Rokhlin actions are necessarily outer. + + + \begin{prop}\label{prop:dsi-wtr-preserves-Z} + Let $A$ be a unital, simple, separable, nuclear, $\cZ$-stable C*-algebra. If $G \act^\alpha A$ is an action of a finite group with the weak tracial Rokhlin property. Then $A^\alpha \subseteq A \rtimes_\alpha G$ is $\cZ$-stable. + \begin{proof} + Let $k \in \bN$. By \cite[Theorem 5.6]{HirshbergOrovitz13} $A \rtimes_\alpha G$ is tracially $\cZ$-absorbing, meaning there are tracially large (in the sense of \cite{TomsWhiteWinter15}) c.p.c. order zero maps $\phi: M_k \to (A \rtimes_\alpha G)_\omega \cap (A \rtimes_\alpha G)'$, which can be chosen to be c.p.c. order zero maps $\phi: M_k \to A_\omega \cap (A \rtimes_\alpha G)'$ by the proof of \cite[Lemma 5.5]{HirshbergOrovitz13}. These tracially large c.p.c. order zero maps yield sequences of positive contractions $c_1 = (c_{1n}),\dots,c_k = (c_{kn}) \in A_\omega \cap (A \rtimes_\alpha G)'$ such that if $(e_n) = e := 1 - \sum_i c_i^*c_i$, we have +\begin{equation} + \lim_{n \to \omega} \max_{\tau \in T(A)} \tau(e_n) = 0, \inf_m \lim _{n \to \omega} \min_{\tau \in T(A)} \tau(c_{1n}^m) > 0 +\end{equation} + and $c_ic_j^* = \delta_{ij}c_1^2$. By \cite[Proposition 4.11]{GardellaHirshberg18} (which is much more general, applicable to all countable amenable groups), $A \subseteq A \rtimes_\alpha G$ has equivariant property (SI) since $A$ has property (SI).\footnote{A unital, separable, simple, nuclear, $\cZ$-stable C*-algebra has property (SI) as in \cite{MatuiSato12}} Consequently there exists $s \in A_\omega \cap (A \rtimes_\alpha G)'$ such that $s^*s = 1 - \sum_i c_i^*c_i$ and $c_1s = s$. Altogether, + \begin{itemize} + \item $c_1 \geq 0$; + \item $c_ic_j^* = \delta_{ij}c_1^2$; + \item $s^*s + \sum_i c_i^*c_i = 1$; + \item $c_1s = s$. + \end{itemize} + As mentioned in the proof of $(iv) \Rightarrow (i)$ of \cite{MatuiSato12}, $\cZ_{n,n+1}$ is the universal C*-algebra generated by $n+1$ elements satisfying the above four relations (see \cite[Proposition 5.1]{RordamWinter10} and \cite[Proposition 2.1]{Sato10}), and consequently we have a unital *-homomorphism $\cZ_{n,n+1} \to A_\omega \cap (A \rtimes_\alpha G)'$. Therefore $\cZ \into A_\omega \cap (A \rtimes_\alpha G)'$, giving that the desired inclusion is $\cZ$-stable by Lemma \ref{lem:fixed-point-in-crossed-product-D-stable}. + \end{proof} + \end{prop} + + + \begin{cor}\label{weaktracialRokhlinintermediate} + Let $A$ be a unital, simple, separable, nuclear, $\cZ$-stable C*-algebra and $G \act^\alpha A$ be an action of a finite group with the weak tracial Rokhlin property. There exists an isomorphism $\Phi: A \rtimes_\alpha G \simeq (A \rtimes_\alpha G) \otimes \cZ$ such that whenever $C$ is a unital C*-algebra satisfying either + \begin{enumerate} + \item $A^\alpha \subseteq C \subseteq A$ or + \item $A \subseteq C \subseteq A \rtimes_\alpha G$, + \end{enumerate} + we have $\Phi(C) = C \otimes \cZ$. + \begin{proof} + This results from combining Proposition \ref{prop:dsi-wtr-preserves-Z} together with Theorem \ref{Galois}, making note that this is an outer action. + \end{proof} + \end{cor} + + \subsection{The diagonal inclusion associated to a group action}\label{ssection:dse-diagonal-inclusions} In the von Neumann setting, a certain diagonal inclusion associated to several automor\hyp{}phisms was considered in \cite{Popa89,Kawamuro99,Burstein10}, and they play a role in subfactor theory. Here we consider a unital C*-algebraic inclusion of the same form. + + \begin{defn} + Let $A$ be a C*-algebra, $\alpha_1,\dots,\alpha_n \in \Aut(A)$. The diagonal inclusion associated to $\alpha_1,\dots,\alpha_n$ is +\begin{equation} + B(\alpha_1,\dots,\alpha_n) = \left\{\bigoplus_{i=1}^n \alpha_i(a) \mid a \in A \right\} \subseteq M_n(A). +\end{equation} + \end{defn} + + If $G \act^\alpha A$ is an action of a finite group, we'll write + \begin{equation} + B(\alpha) = \left\{\bigoplus_{g \in G} \alpha_g(a) \mid a \in A\right\} \subseteq M_{|G|}(A). + \end{equation} +We note that a diagonal $B(\alpha) \subseteq M_{|G|}(A)$ is unique up to unitary conjugation (by permutation unitaries). As $\cD$-stability of an inclusion is preserved under unitary conjugation, there is no ambiguity in speaking of $\cD$-stability of the inclusion $B(\alpha) \subseteq M_{|G|}(A)$. + + \begin{prop}\label{diagonalinclusion} + Let $G \act^\alpha A$ be an action of a countable discrete group on a unital, separable C*-algebra. If $G = \langle g_1,\dots,g_n\rangle$, then $A \subseteq A \rtimes_\alpha G$ is $\cD$-stable if and only if +\begin{equation} + B(\id_A,\alpha_{g_1},\dots,\alpha_{g_n}) \subseteq M_{n+1}(A) +\end{equation} + is $\cD$-stable. + \begin{proof} + First suppose that $A \subseteq A \rtimes_\alpha G$ is $\cD$-stable. Let $\cF \subseteq \cD, \cG \subseteq M_{n+1}(A)$ be finite and $\ee > 0$. Let $\cG' \subseteq A$ be the set of matrix coefficients of elements of $\cG$, together with the identity of $A$, and let $L := \max\{1,\max_{a \in \cG'}\|a\|\}$. Relabel $\id_A,\alpha_{g_1},\dots,\alpha_{g_n}$ as $\alpha_1,\dots,\alpha_{n+1}$. Let +\begin{equation} + \delta := \frac{\ee}{(4L+1)(n+1)^2} +\end{equation} + and let $\psi: \cD \to A$ be a u.c.p. $(\cF,\delta,\cG' \cup\{u_{g_i}\}_{i=1}^n)$-embedding, where $(u_g)$ are the implementing unitaries for $\alpha$. Let $\phi: D \to B(\alpha) \subseteq M_{|G|}(A)$ be given by + \begin{equation} + \phi(d) := \bigoplus_{i=1}^{n+1} (\alpha_i \circ \psi)(d). + \end{equation} + Clearly $\phi$ will be $(\cF,\delta)$-multiplicative since each component is the composition of a *-homomorphism (which are contractive) with a map which is $(\cF,\delta)$-multiplicative. Now for $d \in \cF$ and $a = (a_{ij}) \in \cG$, we have +\begin{equation} +\begin{split} \|[\phi(d),(a_{ij})]\| &\leq \sum_{i,j=1}^{n+1}\|\alpha_i(\psi(d))a_{ij} - a_{ij}\alpha_j(\psi(d))\| \\ + &\leq \sum_{i,j=1}^{n+1}\| \alpha_i(\psi(d))a_{ij} - \psi(d)a_{ij}\| \\ + &\ \ \ + \|\psi(d)a_{ij} - a_{ij}\psi(d)\| + \|a_{ij}\psi(d) - a_{ij}\alpha_j(\psi(d))\| \\ + &\leq \sum_{i,j=1}^{n+1}\|a_{ij}\|\left(\|\alpha_i(\psi(d)) - \psi(d)\| + \|\psi(d) - \alpha_j(\psi(d))\|\right) \\ + &\ \ \ + \|[\psi(d),a_{i,j}]\| \\ + &< (n+1)^2(2L(\delta + \delta) + \delta) \\ + &= (n+1)^2(4L+1)\delta = \ee. +\end{split} +\end{equation} + + Conversely if the associated diagonal inclusion is $\cD$-stable we note that if $(x_k) \subseteq B(\id_A,\alpha_{g_1},\dots,\alpha_{g_n})$ is central for $M_{n+1}(A)$, writing + \begin{equation} + x_k = \bigoplus_{i=1}^{n+1}\alpha_i(a_k) + \end{equation} + yields that $(a_k) \subseteq A$ is central for $A$ and is asymptotically fixed by $\alpha_{g_i}, i=1,\dots,n$. In particular if $\cD \into B(\id_A,\alpha_{g_1},\dots,\alpha_{g_n})_\omega \cap (M_{n+1}(A))'$, then $\cD \into A_\omega \cap (A \rtimes_\alpha G)'$. + \end{proof} + \end{prop} + + \begin{cor} + Let $G \act^\alpha A$ be an action of a finite group on a unital, separable C*-algebra. Then $A \subseteq A \rtimes_\alpha G$ is $\cD$-stable if and only if +\begin{equation} + B(\alpha) \subseteq M_{|G|}(A) +\end{equation} + is $\cD$-stable. + \end{cor} + + + \section{Examples}\label{section:dse-examples} + + \subsection{Non-examples}\label{ssection:dse-non-examples} + We first start with some non-examples. Villadsen's C*-algebras with perforation will be useful (see \cite{TomsWinter09} for good exposition). Let $\cQ = \bigotimes_n M_n$ denote the universal UHF C*-algebra. + + \begin{theorem}[\cite{Villadsen98,Toms08}] + There exists a unital, simple, separable, nuclear C*-algebra $C$ satisfying the UCT such that $C \not\simeq C \otimes \cZ$ and $C$ contains the universal UHF algebra unitally. Moreover $C$ is tracial and can be chosen to be AH with +\begin{equation} + (K_0(A),K_0(A)^+,[1]_0,K_1(A)) = (\bQ,\bQ_+,1,0). +\end{equation} + \end{theorem} + + + \begin{cor} + There exists an embedding $\cQ \into \cQ$ which is not $\cZ$-stable. In particular, it is not $\cQ$-stable. + \begin{proof} + Let $C$ be as above. Note that $\cQ \subseteq C$ so we must find an embedding $C \into \cQ$. As $C$ is unital, separable, exact, satisfies the UCT and has a faithful amenable trace (it has traces, and every such trace will be faithful and amenable since $C$ is nuclear and simple) and there is clearly a morphism between $K_0$-groups, \cite[Theorem D]{Schafhauser20} gives an embedding $C \into \cQ$. Conse\hyp{}quently there is an embedding +\begin{equation} + \cQ \into C \into \cQ +\end{equation} + which is not $\cQ$-stable since there is an intermediate C*-algebra $C$ with $C \not\simeq C \otimes \cZ$. + \end{proof} + \end{cor} + + \begin{cor} + There is an embedding $\cZ \into \cQ$ which is not $\cZ$-stable. + \begin{proof} + Take $C$ as above and take the chain of embeddings (noting that $\cQ$ is $\cZ$-stable) +\begin{equation} + \cZ \into \cQ \otimes \cZ \simeq \cQ \into C \into \cQ. +\end{equation} + \end{proof} + \end{cor} + + \begin{cor} + There is an embedding $\cZ \into \cO_2$ which is not $\cZ$-stable. + \begin{proof} + Just take the same embedding as above together with an embedding $\cQ \into \cO_2$. + \end{proof} + \end{cor} + + \begin{remark} + All *-homomorphisms between strongly self-absorbing C*-alge\hyp{}bras are approximately unitarily equivalent by \cite[Corollary 1.12]{TomsWinter07}, or even asymptotically unitarily equivalent by \cite[Theorem 2.2]{DadarlatWinter09}. Therefore $\cD$-stability is not closed under these equivalences (nor homotopy, see \cite[Corollary 3.1]{DadarlatWinter09}). + \end{remark} + + The only method we have used to show that an inclusion is not $\cD$-stable is by finding an intermediate algebra which is not $\cD$-stable. + There are plenty of examples of stably finite C*-algebras with perforation or higher-stable rank (in particular non-$\cZ$-stable C*-algebras \cite{Rordam04}) \cite{Villadsen98, Villadsen99,ElliottVilladsen00,Toms05,HirshbergRordamWinter07,Toms08b,Toms08,TomsWinter09,deLacerdaMortari09,Tikuisis12}. This gives rise to the following two questions. + + \begin{enumerate} + \item Is there a unital inclusion $B \subseteq A$ of separable C*-algebras such that whenever $C$ is such that $B \subseteq C \subseteq A$, we have $C \simeq C \otimes \cD$ but $B \subseteq A$ is not $\cD$-stable? Is $\cD$-stability equivalent to every intermediate C*-algebra being $\cD$-stable? + \item To get non-examples we use stably finite C*-algebras with perforation in between sufficiently regular C*-algebras. Is there a way to do this for purely infinite C*-algebras, or is finiteness the only obstruction? Thus we can ask: if $\cD$ is a purely infinite strongly self-absorbing C*-algebra, is every embedding of $\cD$ into itself $\cD$-stable? More specifically, if $B \subseteq A$ is a unital inclusion of simple, separable, purely infinite C*-algebras, is the inclusion $\cO_{\infty}$-stable? + \end{enumerate} + + Our third question asks if we can get non-examples arising from dynamical systems. + + \begin{enumerate} + \item[3.] Is there a unital, separable $\cD$-stable C*-algebra and a (finite) group action $G \act^\alpha A$ such that $A \rtimes_\alpha G$ is $\cD$-stable, but the inclusion is not? One would need $A \rtimes_\alpha G$ to be $\cD$-stable for non-dynamical reasons. + \end{enumerate} + + + \subsection{Cyclicly permuting tensor powers}\label{ssetcion:dse-cyclic-tensor-powers} + + Here we give a dynamical example to illustrate the discussion in section \ref{section:dse-crossed-products}. In particular, we can look at a consequence of Corollary \ref{autfixedpoint}. + + \begin{example} + Let $p,q \in \bN$ be coprime and consider the $q$th tensor power of the UHF algebra $A = M_{p^{\infty}}^{\otimes q}$. Let us examine the action $\bZ_q \act^\sigma A$ given by cyclically permuting the tensors: +\begin{equation} + \sigma(a_1 \otimes \cdots \otimes a_q) = a_2 \otimes \cdots \otimes a_q \otimes a_1. +\end{equation} + One can prove directly or use \cite{HirshbergOrovitz13} or \cite{AGJP22} in order to conclude that this action has the weak tracial Rokhlin property and consequently that $A^{\sigma} \subseteq A \rtimes_\sigma \bZ_q$ is $\cZ$-stable. + + Alternatively, one can use techniques similar to \cite{HirshbergWinter07} in order to compute the $K$-theory of the fixed point algebra $A^{\sigma}$ to be +\begin{equation} + K_0((M_{p^{\infty}}^{\otimes q})^{\sigma}) \simeq \dlim\left(\bZ^q, \begin{pmatrix} + p + \frac{p^q - p}{q} & \frac{p^q -p}{q} & \cdots & \frac{p^q - p}{q} \\ + \frac{p^q - p}{q} & p + \frac{p^q - p}{q} & \cdots & \frac{p^q - p}{q} \\ + \vdots & \vdots & \ddots & \vdots \\ + \frac{p^q - p}{q} & \frac{p^q - p}{q} & \cdots & p + \frac{p^q - p}{q} + \end{pmatrix} \right), +\end{equation} + from which one can show that $K_0(A^{\sigma})$ is $p$-divisible.Then using the fact that $K_0(A^{\sigma})$ is $p$-divisible and $A^{\sigma}$ is AF, it follows that $A^{\sigma}$ is $M_{p^{\infty}}$-stable. Using Corollary \ref{autfixedpoint}, we then see that $M_{p^{\infty}} \into (A^{\sigma})_\omega \cap A'$. In particular, we have that $A^{\sigma} \subseteq A \rtimes_\sigma \bZ_q$ is $M_{p^{\infty}}$-stable (since clearly if this embedding is fixed by $\bZ_q$, it will commute with the implementing unitaries as well). + \end{example} + + \begin{example} + Following up on the previous example, if we consider the embedding +\begin{equation} + B := \left\{ \begin{pmatrix} + x \\ & \sigma(x) \\ & & \ddots \\ & & & \sigma^{q-1}(x) + \end{pmatrix} \mid x \in M_{p^{\infty}}^{\otimes q} \right\} \subseteq M_q(M_{p^{\infty}}^{\otimes q}) := A, +\end{equation} + then $B \subseteq A$ is $M_{p^{\infty}}$-stable by Proposition \ref{diagonalinclusion}. + \end{example} + + \subsection{The canonical inclusion of the CAR algebra in $\cO_2$}\label{ssection:dse-car-in-O2} + \begin{example} + Let $\cO_2 = C^*(s_1,s_2)$ be the Cuntz algebra generated by two isometries \cite{Cuntz77}, and consider the inclusion +\begin{equation} + M_{2^{\infty}} \simeq \ov{\text{span}}\{s_\mu s_\nu^* \mid |\mu| = |\nu|\} \subseteq \cO_2, +\end{equation} + where for a word $\mu = \{i_1,\dots,i_p\} \in \{1,2\}^p$, $s_\mu = s_{i_1}\cdots s_{i_p}$. This copy of the CAR algebra is precisely the fixed point subalgebra of the gauge action (see \cite{RaeburnBook}). Consider the endomorphism $\lambda: \cO_2 \to \cO_2$ given by +\begin{equation} + \lambda(x) := s_1xs_1^* + s_2xs_2^*. +\end{equation} + We note that a sequence $(x_n)$ is $\omega$-asymptotically central for $\cO_2$ if and only if it is $\omega$-asymptotically fixed by $\lambda$. Indeed, if $(x_n)$ is central, then $\|\lambda(x_n) - x_n\| \to^{n \to \omega} 0$ since $[x_n,s_i] \to 0$ for $i=1,2$. On the other hand if $(x_n)$ is asymptotically fixed by $\lambda$, then the inequalities + \begin{equation} + \begin{split} + \|s_ix_n - x_ns_i\| &= \|s_1x_ns_1^*s_i + s_2x_ns_2^*s_i - x_ns_i\| \leq \|\lambda(x_n) - x_n\|\|s_i\| \\ + \|s_i^*x_n - x_ns_i^*\| &= \|s_i^*x_n - s_i^*s_1s_1^* - s_i^*s_2x_ns_2^*\| \leq \|s_i^*\|\|\lambda(x_n) - x_n\| + \end{split} + \end{equation} + imply that $(x_n)$ is asymptotically central. + + We note that $\lambda|_{M^{2^{\infty}}}$ is the forward tensor shift if we identify $M_{2^{\infty}} = \bigotimes_{\bN} M_2$ (see for example \cite[Section V.4]{DavidsonBook1}). Now \cite{BSKR93} gives an embedding $\xi: M_2 \into (M_{2^{\infty}})$ such that $\lambda_\omega \circ \xi = \xi$. In particular $M_{2^{\infty}} \into (M_{2^{\infty}})_{\omega} \cap \cO_2'$ so that this inclusion is $M_{2^{\infty}}$-stable. + \end{example} + + Thinking of $\cO_2$ as the semigroup crossed product $\cO_2 \simeq M_{2^{\infty}} \rtimes_\lambda \bN$ (see \cite{Rordam95,Rordam21}), any intermediate C*-algebra is automatically CAR stable. Consequently each intermediate subalgebra $M_{2^{\infty}} \rtimes d\bN = C^*(M_{2^{\infty}},s_1^d)$ is $M_{2^{\infty}}$-stable. We can do this all concurrently. + + \begin{cor} + There exists an isomorphism $\Phi: \cO_2 \simeq \cO_2 \otimes M_{2^{\infty}}$ such that +\begin{equation} + \Phi(C^*(M_{2^{\infty}},s_1^d)) \simeq C^*(M_{2^{\infty}},s_1^d) \otimes M_{2^{\infty}} +\end{equation} + for all $d \in \bN$. The same holds if we replace $M_{2^{\infty}}$ by $\cZ$. + \end{cor} + + + Now let us play with some diagonal inclusions associated to powers of the Bernoulli shift $\lambda$ on $\cO_2$ above. This will be similar to what was discussed in Section \ref{ssection:dse-diagonal-inclusions}, except we allow endomorphisms. + + \begin{example} + + Consider, for $n \in \bN$, the diagonal inclusion +\begin{equation} + B_n:= \left\{ \begin{pmatrix} + x \\ & \lambda(x) \\ & & \ddots \\ & & & \lambda^{n-1}(x) + \end{pmatrix} \mid x \in \cO_2 \right\} \subseteq M_n(\cO_2) =: A_n. +\end{equation} + Note that both $A_n$ and $B_n$ are isomorphic to $\cO_2$, and in fact this gives an non-trivial inclusion of $\cO_2$ into itself which is $\cO_2$-stable. This is $\cO_2$-stable since a sequence is asymptotically fixed by $\lambda$ if and only if it asymptotically commutes with the algebra. A similar argument to that of Proposition \ref{diagonalinclusion} will yield that this inclusion is $\cO_2$-stable. + \end{example} + + One can even restrict the diagonal to elements of the CAR algebra $M_{2^{\infty}} \subseteq \cO_2$ sitting as the fixed point subalgebra of the Gauge action as above. + \begin{example} + Consider +\begin{equation} + B_n^{(2)} := \left\{ \begin{pmatrix} + x \\ & \lambda(x) \\ & & \ddots \\ & & & \lambda^{n-1}(x) \end{pmatrix} \mid x \in M_{2^{\infty}} \right\} \subseteq M_n(\cO_2) = A_n. +\end{equation} + This is $M_{2^{\infty}}$-stable for the same reasons as above. This gives another inclusion $M_{2^{\infty}} \simeq B_n^{(2)} \subseteq M_n(\cO_2) \simeq \cO_2$ which is CAR-stable. + \end{example} + + \bibliographystyle{amsalpha} + \bibliography{biblio} + +\end{document} diff --git a/Unitary groups, K-theory and traces/biblio.bib b/Unitary groups, K-theory and traces/biblio.bib new file mode 100644 index 0000000..cfdb098 --- /dev/null +++ b/Unitary groups, K-theory and traces/biblio.bib @@ -0,0 +1,3763 @@ +#### AAAAAA + +# Albert-Muckenhoupt +@article{AlbertMuckenhoupt57, + title={On matrices of trace zeros}, + author={Albert, Abraham A. and Muckenhoupt, Benjamin}, + journal={Michigan Mathematical Journal}, + volume={4}, + number={1}, + pages={1--3}, + year={1957}, + publisher={University of Michigan, Department of Mathematics} +} + +# Akemann-Anderson-Pedersen +@article {AkemannAndersonPedersen86, + AUTHOR = {Akemann, Charles A. and Anderson, Joel and Pedersen, Gert K.}, + TITLE = {Excising states of {C}*-algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {38}, + YEAR = {1986}, + NUMBER = {5}, + PAGES = {1239--1260}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (46L30)}, + MRNUMBER = {869724}, +MRREVIEWER = {J. W. Bunce}, + DOI = {10.4153/CJM-1986-063-7}, + URL = {https://doi.org/10.4153/CJM-1986-063-7}, +} + +# Al-Rawashdeh-Booth-Giordano +@article {AlBoothGiordano12, + AUTHOR = {Al-Rawashdeh, Ahmed and Booth, Andrew and Giordano, Thierry}, + TITLE = {Unitary groups as a complete invariant}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {262}, + YEAR = {2012}, + NUMBER = {11}, + PAGES = {4711--4730}, + ISSN = {0022-1236}, + MRCLASS = {46L35}, + MRNUMBER = {2913684}, + DOI = {10.1016/j.jfa.2012.03.016}, + URL = {https://doi.org/10.1016/j.jfa.2012.03.016}, +} + +# Alekseev-Schmidt-Thom +@article{AlekseevSchmidtThom23, + title={Amenability for unitary groups of {C}*-algebras}, + author={Alekseev, Vadim and Schmidt, Max and Thom, Andreas}, + journal={arXiv:2305.13181}, + year={2023} +} + +# Amini-Golestani-Jamali-Phillips +@article{AGJP22, + title={Group actions on simple tracially $\mathcal{Z}$-absorbing {C}*-algebras}, + author={Amini, Massoud and Golestani, Nasser and Jamali, Saeid and Phillips, N. Christopher}, + journal={arXiv:2204.03615}, + year={2022}, +} + +# Amrutam-Kalantar +@article {AmrutamKalantar20, + AUTHOR = {Amrutam, Tattwamasi and Kalantar, Mehrdad}, + TITLE = {On simplicity of intermediate {C}*-algebras}, + JOURNAL = {Ergodic Theory Dynam. Systems}, + FJOURNAL = {Ergodic Theory and Dynamical Systems}, + VOLUME = {40}, + YEAR = {2020}, + NUMBER = {12}, + PAGES = {3181--3187}, + ISSN = {0143-3857}, + MRCLASS = {46L55 (37A55 46L10 46L35 46L89)}, + MRNUMBER = {4170599}, +MRREVIEWER = {Bruno Brogni Uggioni}, + DOI = {10.1017/etds.2019.34}, + URL = {https://doi.org/10.1017/etds.2019.34}, +} + +# Ando-Haagerup +@article {AndoHaagerup14, + AUTHOR = {Ando, Hiroshi and Haagerup, Uffe}, + TITLE = {Ultraproducts of von {N}eumann algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {266}, + YEAR = {2014}, + NUMBER = {12}, + PAGES = {6842--6913}, + ISSN = {0022-1236}, + MRCLASS = {46M07 (46L10)}, + MRNUMBER = {3198856}, +MRREVIEWER = {E. St{\o}rmer}, + DOI = {10.1016/j.jfa.2014.03.013}, + URL = {https://doi.org/10.1016/j.jfa.2014.03.013}, +} + +# Ara-Mathieu +@book {AraMathieubook, + AUTHOR = {Ara, Pere and Mathieu, Martin}, + TITLE = {Local multipliers of {C}*-algebras}, + SERIES = {Springer Monographs in Mathematics}, + PUBLISHER = {Springer-Verlag London, Ltd., London}, + YEAR = {2003}, + PAGES = {xii+319}, + ISBN = {1-85233-237-9}, + MRCLASS = {46L05 (16S90 46Kxx 46L40 47B47 47L25)}, + MRNUMBER = {1940428}, +MRREVIEWER = {Michael Frank}, + DOI = {10.1007/978-1-4471-0045-4}, + URL = {https://doi.org/10.1007/978-1-4471-0045-4}, +} + +# Argerami-Farenick +@article {ArgeramiFarenick08, + AUTHOR = {Argerami, Mart\'{\i}n and Farenick, Douglas R.}, + TITLE = {Local multiplier algebras, injective envelopes, and type {I} + {W}*-algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {59}, + YEAR = {2008}, + NUMBER = {2}, + PAGES = {237--245}, + ISSN = {0379-4024,1841-7744}, + MRCLASS = {46L05 (46L07)}, + MRNUMBER = {2411044}, +MRREVIEWER = {Pere\ Ara}, +} + +#### BBBBBBBBBBBBBBBBBBBBBBBB + +#Bresar +@article {Bresar93, + AUTHOR = {Bre\v{s}ar, Matej}, + TITLE = {Commuting traces of biadditive mappings, + commutativity-preserving mappings and {L}ie mappings}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {335}, + YEAR = {1993}, + NUMBER = {2}, + PAGES = {525--546}, + ISSN = {0002-9947}, + MRCLASS = {16W25 (16N60 16W10)}, + MRNUMBER = {1069746}, +MRREVIEWER = {Howard E. Bell}, + DOI = {10.2307/2154392}, + URL = {https://doi.org/10.2307/2154392}, +} + +#Bisch +@article {Bisch90, + AUTHOR = {Bisch, Dietmar H.}, + TITLE = {On the existence of central sequences in subfactors}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {321}, + YEAR = {1990}, + NUMBER = {1}, + PAGES = {117--128}, + ISSN = {0002-9947}, + MRCLASS = {46L35}, + MRNUMBER = {1005075}, +MRREVIEWER = {V. S. Sunder}, + DOI = {10.2307/2001593}, + URL = {https://doi.org/10.2307/2001593}, +} + +@article {Bisch94, + AUTHOR = {Bisch, Dietmar H.}, + TITLE = {Central sequences in subfactors. {II}}, + JOURNAL = {Proc. Amer. Math. Soc.}, + FJOURNAL = {Proceedings of the American Mathematical Society}, + VOLUME = {121}, + YEAR = {1994}, + NUMBER = {3}, + PAGES = {725--731}, + ISSN = {0002-9939}, + MRCLASS = {46L37}, + MRNUMBER = {1209417}, +MRREVIEWER = {V. S. Sunder}, + DOI = {10.2307/2160268}, + URL = {https://doi.org/10.2307/2160268}, +} + +# Blackadar +@article {Blackadar90, + AUTHOR = {Blackadar, Bruce}, + TITLE = {Symmetries of the {CAR} algebra}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {131}, + YEAR = {1990}, + NUMBER = {3}, + PAGES = {589--623}, + ISSN = {0003-486X}, + MRCLASS = {46L80 (19K14)}, + MRNUMBER = {1053492}, +MRREVIEWER = {Bola O. Balogun}, + DOI = {10.2307/1971472}, + URL = {https://doi.org/10.2307/1971472}, +} + +@book {BlackadarBook, + AUTHOR = {Blackadar, Bruce}, + TITLE = {Operator algebras}, + SERIES = {Encyclopaedia of Mathematical Sciences}, + VOLUME = {122}, + NOTE = {Theory of {C}*-algebras and von {N}eumann algebras, + Operator Algebras and Non-commutative Geometry, {III}}, + PUBLISHER = {Springer-Verlag, Berlin}, + YEAR = {2006}, + PAGES = {xx+517}, + ISBN = {978-3-540-28486-4; 3-540-28486-9}, + MRCLASS = {46L05 (46L10 46L80)}, + MRNUMBER = {2188261}, +MRREVIEWER = {Paul Jolissaint}, + DOI = {10.1007/3-540-28517-2}, + URL = {https://doi.org/10.1007/3-540-28517-2}, +} + +# Blacakdar-Kumjian-Rordam +@article {BKR92, + AUTHOR = {Blackadar, Bruce and Kumjian, Alexander and R{\o}rdam, Mikael}, + TITLE = {Approximately central matrix units and the structure of + noncommutative tori}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {6}, + YEAR = {1992}, + NUMBER = {3}, + PAGES = {267--284}, + ISSN = {0920-3036}, + MRCLASS = {46L87 (46L05)}, + MRNUMBER = {1189278}, +MRREVIEWER = {Chi Wai Leung}, + DOI = {10.1007/BF00961466}, + URL = {https://doi.org/10.1007/BF00961466}, +} + +# Booth +@mastersthesis{Booth98, + title={The unitary group as a complete invariant for simple unital {AF} algebras}, + author={Booth, Andrew}, + year={1998}, + school={University of Ottawa (Canada)}, +} + +# Bosa-Gabe-Sims-White +@article {BGSW22, + AUTHOR = {Bosa, Joan and Gabe, James and Sims, Aidan and White, Stuart}, + TITLE = {The nuclear dimension of $\mathcal{O}_\infty$-stable {C}*-algebras}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {401}, + YEAR = {2022}, + PAGES = {Paper No. 108250, 51}, + ISSN = {0001-8708}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {4392219}, + DOI = {10.1016/j.aim.2022.108250}, + URL = {https://doi.org/10.1016/j.aim.2022.108250}, +} + +# Brattelo-Stormer-Kishimoto-Rordam +@article {BSKR93, + AUTHOR = {Bratteli, Ola and St{\o}rmer, Erling and Kishimoto, Akitaka and + R{\o}rdam, Mikael}, + TITLE = {The crossed product of a {UHF} algebra by a shift}, + JOURNAL = {Ergodic Theory Dynam. Systems}, + FJOURNAL = {Ergodic Theory and Dynamical Systems}, + VOLUME = {13}, + YEAR = {1993}, + NUMBER = {4}, + PAGES = {615--626}, + ISSN = {0143-3857}, + MRCLASS = {46L55 (46L35)}, + MRNUMBER = {1257025}, +MRREVIEWER = {Robert J. Archbold}, + DOI = {10.1017/S0143385700007574}, + URL = {https://doi.org/10.1017/S0143385700007574}, +} + +# Brodzku-Cave-Li +@article {BrodzkiCaveLi17, + AUTHOR = {Brodzki, Jacek and Cave, Chris and Li, Kang}, + TITLE = {Exactness of locally compact groups}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {312}, + YEAR = {2017}, + PAGES = {209--233}, + ISSN = {0001-8708,1090-2082}, + MRCLASS = {46L10 (19K56 22D25)}, + MRNUMBER = {3635811}, +MRREVIEWER = {Judith\ A.\ Packer}, + DOI = {10.1016/j.aim.2017.03.020}, + URL = {https://doi.org/10.1016/j.aim.2017.03.020}, +} + +# Brown +@article {Brown81, + AUTHOR = {Brown, Lawrence G.}, + TITLE = {Ext of certain free product {C}*-algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {6}, + YEAR = {1981}, + NUMBER = {1}, + PAGES = {135--141}, + ISSN = {0379-4024}, + MRCLASS = {46M20 (46L05)}, + MRNUMBER = {637007}, +MRREVIEWER = {Claude\ Schochet}, +} + +# Brown-Ozawa +@book {BrownOzawa, + AUTHOR = {Brown, Nathanial P. and Ozawa, Narutaka}, + TITLE = {{C}*-algebras and finite-dimensional approximations}, + SERIES = {Graduate Studies in Mathematics}, + VOLUME = {88}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {2008}, + PAGES = {xvi+509}, + ISBN = {978-0-8218-4381-9; 0-8218-4381-8}, + MRCLASS = {46L05 (43A07 46-02 46L10)}, + MRNUMBER = {2391387}, +MRREVIEWER = {Mikael R{\o}rdam}, + DOI = {10.1090/gsm/088}, + URL = {https://doi.org/10.1090/gsm/088}, +} + +# Bunce +@article {Bunce72, + AUTHOR = {Bunce, John}, + TITLE = {Characterizations of amenable and strongly amenable {C}*-algebras}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {43}, + YEAR = {1972}, + PAGES = {563--572}, + ISSN = {0030-8730}, + MRCLASS = {46L05}, + MRNUMBER = {320764}, +MRREVIEWER = {B. E. Johnson}, + URL = {http://projecteuclid.org/euclid.pjm/1102959351}, +} + +# Burnstein +@article {Burstein10, + AUTHOR = {Burstein, Richard D.}, + TITLE = {Commuting square subfactors and central sequences}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {21}, + YEAR = {2010}, + NUMBER = {1}, + PAGES = {117--131}, + ISSN = {0129-167X}, + MRCLASS = {46L37}, + MRNUMBER = {2642989}, +MRREVIEWER = {Toshihiko Masuda}, + DOI = {10.1142/S0129167X10005945}, + URL = {https://doi.org/10.1142/S0129167X10005945}, +} + +### CCCCCCCCCCCCCCCCCCCCCC + +# Cameron-Smith +@article {CameronSmith19, + AUTHOR = {Cameron, Jan and Smith, Roger R.}, + TITLE = {A {G}alois correspondence for reduced crossed products of + simple {C}*-algebras by discrete groups}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {71}, + YEAR = {2019}, + NUMBER = {5}, + PAGES = {1103--1125}, + ISSN = {0008-414X}, + MRCLASS = {46L55 (46L40)}, + MRNUMBER = {4010423}, +MRREVIEWER = {Robert S. Doran}, + DOI = {10.4153/cjm-2018-014-6}, + URL = {https://doi.org/10.4153/cjm-2018-014-6}, +} + +# Carrion-Castillejos-Evington-Gabe-Schafhauser-Tikuisis-White +@misc{CCEGSTW, + title={Tracially complete {C}*-algebras}, + author={Carri{\'o}n, Jos{\'e} and Castillejos, Jorge and Evington, Samuel and Gabe, James and Schafhauser, Christopher and Tikuisis, Aaron and White, Stuart}, + note={Manuscript in preparation.} +} + +#Carrion-Gabe-Schafhauser-Tikuisis-White +@article{CGSTW23, + title={Classifying *-homomorphisms {I}: Unital simple nuclear {C}*-algebras}, + author={Carri{\'o}n, Jos{\'e} R. and Gabe, James and Schafhauser, Christopher and Tikuisis, Aaron, and White, Stuart}, + journal={arXiv:2307.06480}, + year={2023} +} + +# Castillejos-Evington-Tikuisis-White-Winter +@article {CETWW21, + AUTHOR = {Castillejos, Jorge and Evington, Samuel and Tikuisis, Aaron + and White, Stuart and Winter, Wilhelm}, + TITLE = {Nuclear dimension of simple {C}*-algebras}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {224}, + YEAR = {2021}, + NUMBER = {1}, + PAGES = {245--290}, + ISSN = {0020-9910}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {4228503}, +MRREVIEWER = {Changguo Wei}, + DOI = {10.1007/s00222-020-01013-1}, + URL = {https://doi.org/10.1007/s00222-020-01013-1}, +} + +# Castillejos-Evington-Tikuisis-White +@article{CETW19, + title={Uniform property {G}amma}, + author={Castillejos, Jorge and Evington, Samuel and Tikuisis, Aaron and White, Stuart}, + journal={arXiv:1912.04207}, + year={2019} +} + +@article {CETW22, + AUTHOR = {Castillejos, Jorge and Evington, Samuel and Tikuisis, Aaron + and White, Stuart}, + TITLE = {Uniform property {$\Gamma$}}, + JOURNAL = {Int. Math. Res. Not. IMRN}, + FJOURNAL = {International Mathematics Research Notices. IMRN}, + YEAR = {2022}, + NUMBER = {13}, + PAGES = {9864--9908}, + ISSN = {1073-7928}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {4447140}, + DOI = {10.1093/imrn/rnaa282}, + URL = {https://doi.org/10.1093/imrn/rnaa282}, +} + +# Chand-Robert +@article {ChandRobert23, + AUTHOR = {Chand, Abhinav and Robert, Leonel}, + TITLE = {Simplicity, bounded normal generation, and automatic + continuity of groups of unitaries}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {415}, + YEAR = {2023}, + PAGES = {Paper No. 108894, 52}, + ISSN = {0001-8708}, + MRCLASS = {46L05 (22E65 46H40)}, + MRNUMBER = {4543451}, + DOI = {10.1016/j.aim.2023.108894}, + URL = {https://doi.org/10.1016/j.aim.2023.108894}, +} + +# Choi-Effros +@article {ChoiEffros76, + AUTHOR = {Choi, Man Duen and Effros, Edward G.}, + TITLE = {The completely positive lifting problem for + {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {104}, + YEAR = {1976}, + NUMBER = {3}, + PAGES = {585--609}, + ISSN = {0003-486X}, + MRCLASS = {46L05}, + MRNUMBER = {417795}, +MRREVIEWER = {Maurice J. Dupr\'{e}}, + DOI = {10.2307/1970968}, + URL = {https://doi.org/10.2307/1970968}, +} + +# Choi-Farah-Ozawa +@article {ChoiFarahOzawa14, + AUTHOR = {Choi, Yemon and Farah, Ilijas and Ozawa, Narutaka}, + TITLE = {A nonseparable amenable operator algebra which is not + isomorphic to a {C}*-algebra}, + JOURNAL = {Forum Math. Sigma}, + FJOURNAL = {Forum of Mathematics. Sigma}, + VOLUME = {2}, + YEAR = {2014}, + PAGES = {Paper No. e2, 12}, + ISSN = {2050-5094}, + MRCLASS = {47L30 (03E75 46L05)}, + MRNUMBER = {3177805}, + DOI = {10.1017/fms.2013.6}, + URL = {https://doi.org/10.1017/fms.2013.6}, +} + + +@article {MR3177805, + AUTHOR = {Choi, Yemon and Farah, Ilijas and Ozawa, Narutaka}, + TITLE = {A nonseparable amenable operator algebra which is not + isomorphic to a {${\rm C}^*$}-algebra}, + JOURNAL = {Forum Math. Sigma}, + FJOURNAL = {Forum of Mathematics. Sigma}, + VOLUME = {2}, + YEAR = {2014}, + PAGES = {Paper No. e2, 12}, + ISSN = {2050-5094}, + MRCLASS = {47L30 (03E75 46L05)}, + MRNUMBER = {3177805}, + DOI = {10.1017/fms.2013.6}, + URL = {https://doi.org/10.1017/fms.2013.6}, +} + + +# Connes +@article {Connes75auto, + AUTHOR = {Connes, Alain}, + TITLE = {Classification of automorphisms of hyperfinite factors of type {II}$_1$ and {II}$_{\infty}$ and application to type {III} factors}, + JOURNAL = {Bull. Amer. Math. Soc.}, + FJOURNAL = {Bulletin of the American Mathematical Society}, + VOLUME = {81}, + YEAR = {1975}, + NUMBER = {6}, + PAGES = {1090--1092}, + ISSN = {0002-9904}, + MRCLASS = {46L10}, + MRNUMBER = {388117}, +MRREVIEWER = {Hisashi Choda}, + DOI = {10.1090/S0002-9904-1975-13929-2}, + URL = {https://doi.org/10.1090/S0002-9904-1975-13929-2}, +} + +@article {Connes75anti, + AUTHOR = {Connes, Alain}, + TITLE = {A factor not anti-isomorphic to itself}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {101}, + YEAR = {1975}, + PAGES = {536--554}, + ISSN = {0003-486X}, + MRCLASS = {46L10}, + MRNUMBER = {370209}, +MRREVIEWER = {E. St{\o}rmer}, + DOI = {10.2307/1970940}, + URL = {https://doi.org/10.2307/1970940}, +} + +@article {Connes75Outer, + AUTHOR = {Connes, Alain}, + TITLE = {Outer conjugacy classes of automorphisms of factors}, + JOURNAL = {Ann. Sci. \'{E}cole Norm. Sup. (4)}, + FJOURNAL = {Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure. + Quatri\`eme S\'{e}rie}, + VOLUME = {8}, + YEAR = {1975}, + NUMBER = {3}, + PAGES = {383--419}, + ISSN = {0012-9593}, + MRCLASS = {46L10}, + MRNUMBER = {394228}, +MRREVIEWER = {Hisashi\ Choda}, + URL = {http://www.numdam.org/item?id=ASENS_1975_4_8_3_383_0}, +} + +@article {Connes76, + AUTHOR = {Connes, Alain}, + TITLE = {Classification of injective factors Cases {II}$_1$, {II}$_\infty$, {III}$_\lambda$, $\lambda \neq 1$}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {104}, + YEAR = {1976}, + NUMBER = {1}, + PAGES = {73--115}, + ISSN = {0003-486X}, + MRCLASS = {46L10}, + MRNUMBER = {454659}, +MRREVIEWER = {Fran\c{c}ois Combes}, + DOI = {10.2307/1971057}, + URL = {https://doi.org/10.2307/1971057}, +} + + +@article {Connes77, + AUTHOR = {Connes, Alain}, + TITLE = {Periodic automorphisms of the hyperfinite factor of type {II}$_1$}, + JOURNAL = {Acta Sci. Math. (Szeged)}, + FJOURNAL = {Acta Universitatis Szegediensis. Acta Scientiarum + Mathematicarum}, + VOLUME = {39}, + YEAR = {1977}, + NUMBER = {1-2}, + PAGES = {39--66}, + ISSN = {0001-6969}, + MRCLASS = {46L10}, + MRNUMBER = {448101}, +MRREVIEWER = {Yoshinori Haga}, +} + +@article {Connes78, + AUTHOR = {Connes, Alain}, + TITLE = {On the cohomology of operator algebras}, + JOURNAL = {J. Functional Analysis}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {28}, + YEAR = {1978}, + NUMBER = {2}, + PAGES = {248--253}, + ISSN = {0022-1236}, + MRCLASS = {46L10}, + MRNUMBER = {493383}, +MRREVIEWER = {Man-Duen\ Choi}, + DOI = {10.1016/0022-1236(78)90088-5}, + URL = {https://doi.org/10.1016/0022-1236(78)90088-5}, +} + +@article {Connes85, + AUTHOR = {Connes, Alain}, + TITLE = {FACTORS OF TYPE {III}$_1$, PROPERTY {L}$_\lambda'$ AND CLOSURE OF INNER AUTOMORPHISMS}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {14}, + YEAR = {1985}, + NUMBER = {1}, + PAGES = {189--211}, + ISSN = {0379-4024}, + MRCLASS = {46L35}, + MRNUMBER = {789385}, +MRREVIEWER = {M. Takesaki}, +} + +# Conway +@book {Conway19, + AUTHOR = {Conway, John B.}, + TITLE = {A course in functional analysis}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {96}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1985}, + PAGES = {xiv+404}, + ISBN = {0-387-96042-2}, + MRCLASS = {46-01 (47-01)}, + MRNUMBER = {768926}, +MRREVIEWER = {Greg Robel}, + DOI = {10.1007/978-1-4757-3828-5}, + URL = {https://doi.org/10.1007/978-1-4757-3828-5}, +} + +# Cuntz +@article {Cuntz77, + AUTHOR = {Cuntz, Joachim}, + TITLE = {Simple {C}*-algebras generated by isometries}, + JOURNAL = {Comm. Math. Phys.}, + FJOURNAL = {Communications in Mathematical Physics}, + VOLUME = {57}, + YEAR = {1977}, + NUMBER = {2}, + PAGES = {173--185}, + ISSN = {0010-3616}, + MRCLASS = {46L05}, + MRNUMBER = {467330}, +MRREVIEWER = {E. St{\o}rmer}, + URL = {http://projecteuclid.org/euclid.cmp/1103901288}, +} + +@article {Cuntz81, + AUTHOR = {Cuntz, Joachim}, + TITLE = {{K}-theory for certain {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {113}, + YEAR = {1981}, + NUMBER = {1}, + PAGES = {181--197}, + ISSN = {0003-486X}, + MRCLASS = {46L05 (16A54 18G99 46M20 58G12)}, + MRNUMBER = {604046}, +MRREVIEWER = {Vern Paulsen}, + DOI = {10.2307/1971137}, + URL = {https://doi.org/10.2307/1971137}, +} + +# Cuntz-Pedersen +@article{CuntzPedersen79, + title={Equivalence and traces on {C}*-algebras}, + author={Cuntz, Joachim and Pedersen, Gert K.}, + journal={Journal of Functional Analysis}, + volume={33}, + number={2}, + pages={135--164}, + year={1979}, + publisher={Elsevier} +} +@article {CuntzPedersen79, + AUTHOR = {Cuntz, Joachim and Pedersen, Gert Kjaerg\.{a}rd}, + TITLE = {Equivalence and traces on {C}*-algebras}, + JOURNAL = {J. Functional Analysis}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {33}, + YEAR = {1979}, + NUMBER = {2}, + PAGES = {135--164}, + ISSN = {0022-1236}, + MRCLASS = {46L05}, + MRNUMBER = {546503}, +MRREVIEWER = {Richard I. Loebl}, + DOI = {10.1016/0022-1236(79)90108-3}, + URL = {https://doi.org/10.1016/0022-1236(79)90108-3}, +} + +### DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD +# Dadarlat +@article {Dadarlat95, + AUTHOR = {Dadarlat, Marius}, + TITLE = {Reduction to dimension three of local spectra of real rank + zero {C}*-algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {460}, + YEAR = {1995}, + PAGES = {189--212}, + ISSN = {0075-4102}, + MRCLASS = {46L85 (19K14 46L35)}, + MRNUMBER = {1316577}, +MRREVIEWER = {Terry A. Loring}, + DOI = {10.1515/crll.1995.460.189}, + URL = {https://doi.org/10.1515/crll.1995.460.189}, +} + +# Dadarlat-Loring +@article {DadarlatLoring96, + AUTHOR = {Dadarlat, Marius and Loring, Terry A.}, + TITLE = {A universal multicoefficient theorem for the {K}asparov + groups}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {84}, + YEAR = {1996}, + NUMBER = {2}, + PAGES = {355--377}, + ISSN = {0012-7094}, + MRCLASS = {46L80 (19K35 46L35)}, + MRNUMBER = {1404333}, +MRREVIEWER = {Kevin McClanahan}, + DOI = {10.1215/S0012-7094-96-08412-4}, + URL = {https://doi.org/10.1215/S0012-7094-96-08412-4}, +} + +@article {DadarlatWinter09, + AUTHOR = {Dadarlat, Marius and Winter, Wilhelm}, + TITLE = {On the {KK}-theory of strongly self-absorbing + {C}*-algebras}, + JOURNAL = {Math. Scand.}, + FJOURNAL = {Mathematica Scandinavica}, + VOLUME = {104}, + YEAR = {2009}, + NUMBER = {1}, + PAGES = {95--107}, + ISSN = {0025-5521}, + MRCLASS = {46L80 (19K35 46L05)}, + MRNUMBER = {2498373}, +MRREVIEWER = {Vladimir Manuilov}, + DOI = {10.7146/math.scand.a-15086}, + URL = {https://doi.org/10.7146/math.scand.a-15086}, +} + +# Davidson +@book {DavidsonBook1, + AUTHOR = {Davidson, Kenneth R.}, + TITLE = {{C}*-algebras by example}, + SERIES = {Fields Institute Monographs}, + VOLUME = {6}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {1996}, + PAGES = {xiv+309}, + ISBN = {0-8218-0599-1}, + MRCLASS = {46Lxx (46-01)}, + MRNUMBER = {1402012}, +MRREVIEWER = {Robert S. Doran}, + DOI = {10.1090/fim/006}, + URL = {https://doi.org/10.1090/fim/006}, +} + +# Davidson-Kennedy +@article{DavidsonKennedy19, + title={Noncommutative choquet theory}, + author={Davidson, Kenneth R and Kennedy, Matthew}, + journal={arXiv:1905.08436}, + year={2019} +} + + + +#de Lacerda Mortari +@book {deLacerdaMortari09, + AUTHOR = {Mortari, Fernando de Lacerda}, + TITLE = {Tracial state spaces of higher stable rank simple + {C}*-algebras}, + NOTE = {Thesis (Ph.D.)--University of Toronto (Canada)}, + PUBLISHER = {ProQuest LLC, Ann Arbor, MI}, + YEAR = {2009}, + PAGES = {41}, + ISBN = {978-0494-61035-0}, + MRCLASS = {Thesis}, + MRNUMBER = {2753146}, + URL = + {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:NR61035}, +} + +# de la Harpe +@incollection {dlHarpe79, + AUTHOR = {de la Harpe, Pierre}, + TITLE = {Moyennabilit\'{e} du groupe unitaire et propri\'{e}t\'{e} + {$P$} de {S}chwartz des alg\`ebres de von {N}eumann}, + BOOKTITLE = {Alg\`ebres d'op\'{e}rateurs ({S}\'{e}m., {L}es + {P}lans-sur-{B}ex, 1978)}, + SERIES = {Lecture Notes in Math.}, + VOLUME = {725}, + PAGES = {220--227}, + PUBLISHER = {Springer, Berlin}, + YEAR = {1979}, + ISBN = {3-540-09512-8}, + MRCLASS = {46L35 (22E65 58B25)}, + MRNUMBER = {548116}, +MRREVIEWER = {Marie\ Choda}, +} + +@article {dlHarpe13, + AUTHOR = {de la Harpe, Pierre}, + TITLE = {Fuglede--{K}adison determinant: theme and variations}, + JOURNAL = {Proc. Natl. Acad. Sci. USA}, + FJOURNAL = {Proceedings of the National Academy of Sciences of the United + States of America}, + VOLUME = {110}, + YEAR = {2013}, + NUMBER = {40}, + PAGES = {15864--15877}, + ISSN = {0027-8424}, + MRCLASS = {46L10}, + MRNUMBER = {3363445}, + DOI = {10.1073/pnas.1202059110}, + URL = {https://doi.org/10.1073/pnas.1202059110}, +} + +# de la Harpe-Skandalis +@article {dlHS84a, + AUTHOR = {de la Harpe, Pierre and Skandalis, Georges}, + TITLE = {D\'{e}terminant associ\'{e} \`a une trace sur une alg\'{e}bre de {B}anach}, + JOURNAL = {Ann. Inst. Fourier (Grenoble)}, + FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier}, + VOLUME = {34}, + YEAR = {1984}, + NUMBER = {1}, + PAGES = {241--260}, + ISSN = {0373-0956}, + MRCLASS = {46L80 (18F25 19K14 19K56 46L05 58G12)}, + MRNUMBER = {743629}, +MRREVIEWER = {G. A. Elliott}, + URL = {http://www.numdam.org/item?id=AIF_1984__34_1_241_0}, +} + +@article {dlHS84b, + AUTHOR = {de la Harpe, Pierre and Skandalis, Georges}, + TITLE = {Produits finis de commutateurs dans les {C}*-alg\`ebres}, + JOURNAL = {Ann. Inst. Fourier (Grenoble)}, + FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier}, + VOLUME = {34}, + YEAR = {1984}, + NUMBER = {4}, + PAGES = {169--202}, + ISSN = {0373-0956}, + MRCLASS = {46L05 (18F25 19B99 19C09 19K14 46L80 58G12)}, + MRNUMBER = {766279}, +MRREVIEWER = {G. A. Elliott}, + URL = {http://www.numdam.org/item?id=AIF_1984__34_4_169_0}, +} + +# Dummit-Foote +@book {DummitFoote, + AUTHOR = {Dummit, David S. and Foote, Richard M.}, + TITLE = {Abstract algebra}, + EDITION = {Third}, + PUBLISHER = {John Wiley \& Sons, Inc., Hoboken, NJ}, + YEAR = {2004}, + PAGES = {xii+932}, + ISBN = {0-471-43334-9}, + MRCLASS = {00-01 (16-01 20-01)}, + MRNUMBER = {2286236}, +} + +# Dye +@article {Dye53, + AUTHOR = {Dye, H. A.}, + TITLE = {The unitary structure in finite rings of operators}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {20}, + YEAR = {1953}, + PAGES = {55--69}, + ISSN = {0012-7094}, + MRCLASS = {46.3X}, + MRNUMBER = {52695}, +MRREVIEWER = {F. I. Mautner}, + URL = {http://projecteuclid.org/euclid.dmj/1077465064}, +} + +@article {Dye55, + AUTHOR = {Dye, H. A.}, + TITLE = {On the geometry of projections in certain operator algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {61}, + YEAR = {1955}, + PAGES = {73--89}, + ISSN = {0003-486X}, + MRCLASS = {46.0X}, + MRNUMBER = {66568}, +MRREVIEWER = {J. Dixmier}, + DOI = {10.2307/1969620}, + URL = {https://doi.org/10.2307/1969620}, +} + +### EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE +# Elliott +@article {Elliott76, + AUTHOR = {Elliott, George A.}, + TITLE = {On the classification of inductive limits of sequences of + semisimple finite-dimensional algebras}, + JOURNAL = {J. Algebra}, + FJOURNAL = {Journal of Algebra}, + VOLUME = {38}, + YEAR = {1976}, + NUMBER = {1}, + PAGES = {29--44}, + ISSN = {0021-8693}, + MRCLASS = {46L05 (16A46)}, + MRNUMBER = {397420}, +MRREVIEWER = {Horst Behncke}, + DOI = {10.1016/0021-8693(76)90242-8}, + URL = {https://doi.org/10.1016/0021-8693(76)90242-8}, +} + + +@incollection {Elliott93a, + AUTHOR = {Elliott, George A.}, + TITLE = {A classification of certain simple {$C^*$}-algebras}, + BOOKTITLE = {Quantum and non-commutative analysis ({K}yoto, 1992)}, + SERIES = {Math. Phys. Stud.}, + VOLUME = {16}, + PAGES = {373--385}, + PUBLISHER = {Kluwer Acad. Publ., Dordrecht}, + YEAR = {1993}, + ISBN = {0-7923-2532-X}, + MRCLASS = {46L35 (19K14 46L05 46L80)}, + MRNUMBER = {1276305}, +MRREVIEWER = {Terry\ A.\ Loring}, +} + +@article {Elliott93, + AUTHOR = {Elliott, George A.}, + TITLE = {On the classification of {C}*-algebras of real rank zero}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {443}, + YEAR = {1993}, + PAGES = {179--219}, + ISSN = {0075-4102}, + MRCLASS = {46L05 (19K14 46L80)}, + MRNUMBER = {1241132}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1515/crll.1993.443.179}, + URL = {https://doi.org/10.1515/crll.1993.443.179}, +} + +@article{Elliott22, +title="K-Theory and Traces", +author="Elliott, George A.", +journal="C. R. Math. Rep. Acad. Sci. Canada", +volume="44", +number="1", +pages="1--15", +year="2022" +} + +# Elliott-Gong-Lin-Niu +@article{EGLN15, + title={On the classification of simple amenable {C}*-algebras with finite decomposition rank, {II}}, + author={Elliott, George A. and Gong, Guihua and Lin, H. and Niu, Zhuang}, + journal={arXiv:1507.03437}, + year={2015} +} + +# Elliott-Li-Niu +#@article{ElliottLiNiu22, +# title={A remark on {V}illadsen algebras}, +# author={Elliott, George A. and Li, Chun G. and Niu, Zhuang}, +# journal={arXiv:2209.10649}, +# year={2022} +#} + +@misc{ElliottLiNiu23, + title={Remarks on {V}illadsen algebras}, + author={Elliott, George A. and Li, Chun G. and Niu, Zhuang}, + year={2023}, + eprint={2209.10649}, + archivePrefix={arXiv}, + primaryClass={math.OA}, + NOTE={preprint} +} + +# Elliott-Niu +@article {ElliottNiu12, + AUTHOR = {Elliott, George A. and Niu, Zhuang}, + TITLE = {Extended rotation algebras: adjoining spectral projections to + rotation algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {665}, + YEAR = {2012}, + PAGES = {1--71}, + ISSN = {0075-4102}, + MRCLASS = {46L05}, + MRNUMBER = {2908740}, +MRREVIEWER = {William Paschke}, + DOI = {10.1515/CRELLE.2011.112}, + URL = {https://doi.org/10.1515/CRELLE.2011.112}, +} + +@article {ElliottNiu15, + AUTHOR = {Elliott, George A. and Niu, Zhuang}, + TITLE = {All irrational extended rotation algebras are {AF} algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {67}, + YEAR = {2015}, + NUMBER = {4}, + PAGES = {810--826}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {3361014}, +MRREVIEWER = {Katsutoshi Kawashima}, + DOI = {10.4153/CJM-2014-022-5}, + URL = {https://doi.org/10.4153/CJM-2014-022-5}, +} + +@incollection {ElliottNiu16, + AUTHOR = {Elliott, George A. and Niu, Zhuang}, + TITLE = {On the classification of simple amenable {C}*-algebras with + finite decomposition rank}, + BOOKTITLE = {Operator algebras and their applications}, + SERIES = {Contemp. Math.}, + VOLUME = {671}, + PAGES = {117--125}, + PUBLISHER = {Amer. Math. Soc., Providence, RI}, + YEAR = {2016}, + MRCLASS = {46L35}, + MRNUMBER = {3546681}, + DOI = {10.1090/conm/671/13506}, + URL = {https://doi.org/10.1090/conm/671/13506}, +} + +# Elliott-Villadsen +@article {ElliottVilladsen00, + AUTHOR = {Elliott, George A. and Villadsen, Jesper}, + TITLE = {Perforated ordered {$K_0$}-groups}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {52}, + YEAR = {2000}, + NUMBER = {6}, + PAGES = {1164--1191}, + ISSN = {0008-414X}, + MRCLASS = {46L80 (19K14 46L05)}, + MRNUMBER = {1794301}, + DOI = {10.4153/CJM-2000-049-9}, + URL = {https://doi.org/10.4153/CJM-2000-049-9}, +} + +# Echterhoff-Rordam +@article{EchterhoffRordam21, + title={Inclusions of {C}*-algebras arising from fixed-point algebras}, + author={Echterhoff, Siegfried and R{\o}rdam, Mikael}, + journal={arXiv:2108.08832}, + year={2021} +} + +# Enders-Schemaitat-Tikuisis +@article{EndersSchemaitatTikuisis23, + title={Corrigendum to ``{$K$}-theoretic characterization of {C}*-algebras with approximately inner flip''}, + author={Enders, Dominic and Schemaitat, Andr{\'e} and Tikuisis, Aaron}, + journal={arXiv:2303.11106}, + year={2023} +} + +### FFFFFFFFFFFFFFFFFFFFFF + +#FangGeLi +@article {FangGeLi06, + AUTHOR = {Fang, Junsheng and Ge, Liming and Li, Weihua}, + TITLE = {Central sequence algebras of von {N}eumann algebras}, + JOURNAL = {Taiwanese J. Math.}, + FJOURNAL = {Taiwanese Journal of Mathematics}, + VOLUME = {10}, + YEAR = {2006}, + NUMBER = {1}, + PAGES = {187--200}, + ISSN = {1027-5487}, + MRCLASS = {46L10 (46M07)}, + MRNUMBER = {2186173}, +MRREVIEWER = {H. Halpern}, + DOI = {10.11650/twjm/1500403810}, + URL = {https://doi.org/10.11650/twjm/1500403810}, +} + +# Farah-Hirshberg +@article{FarahHirshberg16, + title={A simple {AF} algebra not isomorphic to its opposite}, + author={Farah, Ilijas and Hirshberg, Ilan}, + journal={arXiv:1612.01170}, + year={2016} +} + +@article {FarahHirshberg17, + AUTHOR = {Farah, Ilijas and Hirshberg, Ilan}, + TITLE = {Simple nuclear {C}*-algebras not isomorphic to their + opposites}, + JOURNAL = {Proc. Natl. Acad. Sci. USA}, + FJOURNAL = {Proceedings of the National Academy of Sciences of the United + States of America}, + VOLUME = {114}, + YEAR = {2017}, + NUMBER = {24}, + PAGES = {6244--6249}, + ISSN = {0027-8424}, + MRCLASS = {46L10}, + MRNUMBER = {3667529}, +MRREVIEWER = {Fyodor A. Sukochev}, + DOI = {10.1073/pnas.1619936114}, + URL = {https://doi.org/10.1073/pnas.1619936114}, +} + +# Folland +@book {Follandbook16, + AUTHOR = {Folland, Gerald B.}, + TITLE = {A course in abstract harmonic analysis}, + SERIES = {Textbooks in Mathematics}, + EDITION = {Second}, + PUBLISHER = {CRC Press, Boca Raton, FL}, + YEAR = {2016}, + PAGES = {xiii+305 pp.+loose errata}, + ISBN = {978-1-4987-2713-6}, + MRCLASS = {43-01 (22-01 42-01 46-01)}, + MRNUMBER = {3444405}, +MRREVIEWER = {D. L. Salinger}, +} + +# Fuglede-Kadison +@article {FugledeKadison52, + AUTHOR = {Fuglede, Bent and Kadison, Richard V.}, + TITLE = {Determinant theory in finite factors}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {55}, + YEAR = {1952}, + PAGES = {520--530}, + ISSN = {0003-486X}, + MRCLASS = {46.3X}, + MRNUMBER = {52696}, +MRREVIEWER = {F.\ I.\ Mautner}, + DOI = {10.2307/1969645}, + URL = {https://doi.org/10.2307/1969645}, +} + +### GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG +# Gabe +@article{Gabe19, + title={Classification of $\mathcal{O}_\infty$-stable {C}*-algebras}, + author={Gabe, James}, + journal={arXiv:1910.06504}, + year={2019} +} + +@article {Gabe20, + AUTHOR = {Gabe, James}, + TITLE = {A new proof of {K}irchberg's $\mathcal{O}_2$-stable classification}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {761}, + YEAR = {2020}, + PAGES = {247--289}, + ISSN = {0075-4102}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {4080250}, +MRREVIEWER = {Francesc Perera}, + DOI = {10.1515/crelle-2018-0010}, + URL = {https://doi.org/10.1515/crelle-2018-0010}, +} + +# Gardella +@article{Gardella17, + title={{R}okhlin-type properties for group actions on {C}*-algebras}, + author={Gardella, Eusebio}, + journal={Lecture Notes, IPM, Tehran}, + year={2017} +} + +@book {GardellaBook, + AUTHOR = {Gardella, Emilio Eusebio}, + TITLE = {Compact group actions on {C}*-algebras: classification, + non-classifiability, and crossed products and rigidity results + for {L}p-operator algebras}, + NOTE = {Thesis (Ph.D.)--University of Oregon}, + PUBLISHER = {ProQuest LLC, Ann Arbor, MI}, + YEAR = {2015}, + PAGES = {713}, + ISBN = {978-1321-96796-8}, + MRCLASS = {Thesis}, + MRNUMBER = {3407494}, + URL = + {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3717348}, +} + +# Gardella-Hirshberg +@article{GardellaHirshberg18, + title={Strongly outer actions of amenable groups on $\mathcal{Z}$-stable {C}*-algebras}, + author={Gardella, Eusebio and Hirshberg, Ilan}, + journal={arXiv:1811.00447}, + year={2018} +} + +# Gardella-Lupini +@article {GardellaLupini18, + AUTHOR = {Gardella, Eusebio and Lupini, Martino}, + TITLE = {Applications of model theory to {C}*-dynamics}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {275}, + YEAR = {2018}, + NUMBER = {7}, + PAGES = {1889--1942}, + ISSN = {0022-1236}, + MRCLASS = {03C98 (28D05 37A55 46L40 46L55 46M07)}, + MRNUMBER = {3832010}, +MRREVIEWER = {Alessandro Vignati}, + DOI = {10.1016/j.jfa.2018.03.020}, + URL = {https://doi.org/10.1016/j.jfa.2018.03.020}, +} + +# Geffen-Ursu +@article{GeffenUrsu23, + title={Simplicity of crossed products by FC-hypercentral groups}, + author={Geffen, Shirly and Ursu, Dan}, + journal={arXiv:2304.07852}, + year={2023} +} + +# Giordano-Pestov +@article {GiordanoPestov02, + AUTHOR = {Giordano, Thierry and Pestov, Vladimir}, + TITLE = {Some extremely amenable groups}, + JOURNAL = {C. R. Math. Acad. Sci. Paris}, + FJOURNAL = {Comptes Rendus Math\'{e}matique. Acad\'{e}mie des Sciences. + Paris}, + VOLUME = {334}, + YEAR = {2002}, + NUMBER = {4}, + PAGES = {273--278}, + ISSN = {1631-073X,1778-3569}, + MRCLASS = {43A07 (46L10)}, + MRNUMBER = {1891002}, + DOI = {10.1016/S1631-073X(02)02218-5}, + URL = {https://doi.org/10.1016/S1631-073X(02)02218-5}, +} + +@article {GiordanoPestov07, + AUTHOR = {Giordano, Thierry and Pestov, Vladimir}, + TITLE = {Some extremely amenable groups related to operator algebras + and ergodic theory}, + JOURNAL = {J. Inst. Math. Jussieu}, + FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. + Journal de l'Institut de Math\'{e}matiques de Jussieu}, + VOLUME = {6}, + YEAR = {2007}, + NUMBER = {2}, + PAGES = {279--315}, + ISSN = {1474-7480,1475-3030}, + MRCLASS = {22D25 (22F50 37A15 43A07 46L99)}, + MRNUMBER = {2311665}, +MRREVIEWER = {Matthew\ D.\ Daws}, + DOI = {10.1017/S1474748006000090}, + URL = {https://doi.org/10.1017/S1474748006000090}, +} + + +# Giordano-Sierakowski +@article {GiordanoSierakowski16, + AUTHOR = {Giordano, Thierry and Sierakowski, Adam}, + TITLE = {The general linear group as a complete invariant for + {C}*-algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {76}, + YEAR = {2016}, + NUMBER = {2}, + PAGES = {249--269}, + ISSN = {0379-4024}, + MRCLASS = {46L35 (47L80)}, + MRNUMBER = {3552377}, +MRREVIEWER = {Marius V. Ionescu}, + DOI = {10.7900/jot.2015may27.2112}, + URL = {https://doi.org/10.7900/jot.2015may27.2112}, +} + +# Giordano-Skandalis +@article {GiordanoSkandalis85, + AUTHOR = {Giordano, T. and Skandalis, G.}, + TITLE = {{K}rieger factors isomorphic to their tensor square and pure + point spectrum flows}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {64}, + YEAR = {1985}, + NUMBER = {2}, + PAGES = {209--226}, + ISSN = {0022-1236}, + MRCLASS = {46L35}, + MRNUMBER = {812392}, +MRREVIEWER = {Michel Hilsum}, + DOI = {10.1016/0022-1236(85)90075-8}, + URL = {https://doi.org/10.1016/0022-1236(85)90075-8}, +} + + +# Gong +@article {Gong97, + AUTHOR = {Gong, Guihua}, + TITLE = {On inductive limits of matrix algebras over higher-dimensional + spaces. {I}, {II}}, + JOURNAL = {Math. Scand.}, + FJOURNAL = {Mathematica Scandinavica}, + VOLUME = {80}, + YEAR = {1997}, + NUMBER = {1}, + PAGES = {41--55, 56--100}, + ISSN = {0025-5521}, + MRCLASS = {46L05 (19K14 46L35 46L80)}, + MRNUMBER = {1466905}, +MRREVIEWER = {Terry A. Loring}, + DOI = {10.7146/math.scand.a-12611}, + URL = {https://doi.org/10.7146/math.scand.a-12611}, +} + +# Gong-Jiang-Su +@article {GongJiangSu00, + AUTHOR = {Gong, Guihua and Jiang, Xinhui and Su, Hongbing}, + TITLE = {Obstructions to $\mathcal{Z}$-stability for unital simple + {$C^*$}-algebras}, + JOURNAL = {Canad. Math. Bull.}, + FJOURNAL = {Canadian Mathematical Bulletin. Bulletin Canadien de + Math\'{e}matiques}, + VOLUME = {43}, + YEAR = {2000}, + NUMBER = {4}, + PAGES = {418--426}, + ISSN = {0008-4395}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {1793944}, +MRREVIEWER = {Takahiro Sudo}, + DOI = {10.4153/CMB-2000-050-1}, + URL = {https://doi.org/10.4153/CMB-2000-050-1}, +} + +# Gong-Lin-Niu +@article {GongLinNiu20I, + AUTHOR = {Gong, Guihua and Lin, Huaxin and Niu, Zhuang}, + TITLE = {A classification of finite simple amenable $\mathcal{Z}$-stable + {C}*-algebras, {I}: {C}*-algebras with generalized + tracial rank one}, + JOURNAL = {C. R. Math. Acad. Sci. Soc. R. Can.}, + FJOURNAL = {Comptes Rendus Math\'{e}matiques de l'Acad\'{e}mie des Sciences. La + Soci\'{e}t\'{e} Royale du Canada. Mathematical Reports of the Academy + of Science. The Royal Society of Canada}, + VOLUME = {42}, + YEAR = {2020}, + NUMBER = {3}, + PAGES = {63--450}, + ISSN = {0706-1994}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {4215379}, +} + +@article {GongLinNiu20II, + AUTHOR = {Gong, Guihua and Lin, Huaxin and Niu, Zhuang}, + TITLE = {A classification of finite simple amenable $\mathcal{Z}$-stable {C}*-algebras, {II}: {C}*-algebras with rational generalized tracial rank one}, + JOURNAL = {C. R. Math. Acad. Sci. Soc. R. Can.}, + FJOURNAL = {Comptes Rendus Math\'{e}matiques de l'Acad\'{e}mie des Sciences. La + Soci\'{e}t\'{e} Royale du Canada. Mathematical Reports of the Academy + of Science. The Royal Society of Canada}, + VOLUME = {42}, + YEAR = {2020}, + NUMBER = {4}, + PAGES = {451--539}, + ISSN = {0706-1994}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {4215380}, +} + +# Gong-Lin-Xue +@article {GongLinXue15, + AUTHOR = {Gong, Guihua and Lin, Huaxin and Xue, Yifeng}, + TITLE = {Determinant rank of {$C^*$}-algebras}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {274}, + YEAR = {2015}, + NUMBER = {2}, + PAGES = {405--436}, + ISSN = {0030-8730}, + MRCLASS = {46L06 (46L35 46L80)}, + MRNUMBER = {3332910}, +MRREVIEWER = {Claire Anantharaman-Delaroche}, + DOI = {10.2140/pjm.2015.274.405}, + URL = {https://doi.org/10.2140/pjm.2015.274.405}, +} + +# Goodearl +@book {Goodearlbook, + AUTHOR = {Goodearl, Kenneth R.}, + TITLE = {Partially ordered abelian groups with interpolation}, + SERIES = {Mathematical Surveys and Monographs}, + VOLUME = {20}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {1986}, + PAGES = {xxii+336}, + ISBN = {0-8218-1520-2}, + MRCLASS = {06F20 (16A54 18F25 19K14 46A55 46L80)}, + MRNUMBER = {845783}, +MRREVIEWER = {G. A. Elliott}, + DOI = {10.1090/surv/020}, + URL = {https://doi.org/10.1090/surv/020}, +} + +# Guentner-Willett-Yu +@article{GuentnerWillettYu16, + title={Dynamical complexity and controlled operator {K}-theory}, + author={Guentner, Erik and Willett, Rufus and Yu, Guoliang}, + journal={arXiv:1609.02093}, + year={2016} +} + +### HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + +# Haagerup +@article {Haagerup79I, + AUTHOR = {Haagerup, Uffe}, + TITLE = {Operator-valued weights in von {N}eumann algebras. {I}}, + JOURNAL = {J. Functional Analysis}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {32}, + YEAR = {1979}, + NUMBER = {2}, + PAGES = {175--206}, + ISSN = {0022-1236}, + MRCLASS = {46L10 (46L50)}, + MRNUMBER = {534673}, +MRREVIEWER = {M.\ Takesaki}, + DOI = {10.1016/0022-1236(79)90053-3}, + URL = {https://doi.org/10.1016/0022-1236(79)90053-3}, +} + +@article {Haagerup79II, + AUTHOR = {Haagerup, Uffe}, + TITLE = {Operator-valued weights in von {N}eumann algebras. {II}}, + JOURNAL = {J. Functional Analysis}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {33}, + YEAR = {1979}, + NUMBER = {3}, + PAGES = {339--361}, + ISSN = {0022-1236}, + MRCLASS = {46L10 (46L50)}, + MRNUMBER = {549119}, +MRREVIEWER = {M.\ Takesaki}, + DOI = {10.1016/0022-1236(79)90072-7}, + URL = {https://doi.org/10.1016/0022-1236(79)90072-7}, +} + +@article {Haagerup83, + AUTHOR = {Haagerup, Uffe}, + TITLE = {All nuclear {C}*-algebras are amenable}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {74}, + YEAR = {1983}, + NUMBER = {2}, + PAGES = {305--319}, + ISSN = {0020-9910,1432-1297}, + MRCLASS = {46L05}, + MRNUMBER = {723220}, +MRREVIEWER = {J.\ W.\ Bunce}, + DOI = {10.1007/BF01394319}, + URL = {https://doi.org/10.1007/BF01394319}, +} + +@article {Haagerup87, + AUTHOR = {Haagerup, Uffe}, + TITLE = {{C}onnes' bicentralizer problem and uniqueness of the injective + factor of type {III}$_1$}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {158}, + YEAR = {1987}, + NUMBER = {1-2}, + PAGES = {95--148}, + ISSN = {0001-5962}, + MRCLASS = {46L35}, + MRNUMBER = {880070}, +MRREVIEWER = {Steve Wright}, + DOI = {10.1007/BF02392257}, + URL = {https://doi.org/10.1007/BF02392257}, +} + +# Hall +@book {HallBook, + AUTHOR = {Hall, Brian}, + TITLE = {Lie groups, {L}ie algebras, and representations}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {222}, + EDITION = {Second}, + NOTE = {An elementary introduction}, + PUBLISHER = {Springer, Cham}, + YEAR = {2015}, + PAGES = {xiv+449}, + ISBN = {978-3-319-13466-6; 978-3-319-13467-3}, + MRCLASS = {22-01 (17-01)}, + MRNUMBER = {3331229}, + DOI = {10.1007/978-3-319-13467-3}, + URL = {https://doi.org/10.1007/978-3-319-13467-3}, +} + + +# Handelman-Rossmann +@article {HandelmanRossmann84, + AUTHOR = {Handelman, David and Rossmann, Wulf}, + TITLE = {Product type actions of finite and compact groups}, + JOURNAL = {Indiana Univ. Math. J.}, + FJOURNAL = {Indiana University Mathematics Journal}, + VOLUME = {33}, + YEAR = {1984}, + NUMBER = {4}, + PAGES = {479--509}, + ISSN = {0022-2518,1943-5258}, + MRCLASS = {46L80 (16A54 18F25 22D25 46L55 46M20)}, + MRNUMBER = {749311}, +MRREVIEWER = {Colin\ E.\ Sutherland}, + DOI = {10.1512/iumj.1984.33.33026}, + URL = {https://doi.org/10.1512/iumj.1984.33.33026}, +} + +@article {HandelmanRossmann85, + AUTHOR = {Handelman, David and Rossmann, Wulf}, + TITLE = {Actions of compact groups on {AF} {C}*-algebras}, + JOURNAL = {Illinois J. Math.}, + FJOURNAL = {Illinois Journal of Mathematics}, + VOLUME = {29}, + YEAR = {1985}, + NUMBER = {1}, + PAGES = {51--95}, + ISSN = {0019-2082,1945-6581}, + MRCLASS = {46L55 (22D25 46L80)}, + MRNUMBER = {769758}, +MRREVIEWER = {Thierry\ Fack}, + URL = {http://projecteuclid.org/euclid.ijm/1256045841}, +} + +# Hatori-Molnar +@article {HatoriMolnar14, + AUTHOR = {Hatori, Osamu and Moln\'{a}r, Lajos}, + TITLE = {Isometries of the unitary groups and {T}hompson isometries of + the spaces of invertible positive elements in + {C}*-algebras}, + JOURNAL = {J. Math. Anal. Appl.}, + FJOURNAL = {Journal of Mathematical Analysis and Applications}, + VOLUME = {409}, + YEAR = {2014}, + NUMBER = {1}, + PAGES = {158--167}, + ISSN = {0022-247X}, + MRCLASS = {46L05}, + MRNUMBER = {3095026}, +MRREVIEWER = {Yangping Jing}, + DOI = {10.1016/j.jmaa.2013.06.065}, + URL = {https://doi.org/10.1016/j.jmaa.2013.06.065}, +} + +# Herman-Ocneanu +@article {HermanOcneanu84, + AUTHOR = {Herman, Richard H. and Ocneanu, Adrian}, + TITLE = {Stability for integer actions on {UHF} {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {59}, + YEAR = {1984}, + NUMBER = {1}, + PAGES = {132--144}, + ISSN = {0022-1236}, + MRCLASS = {46L40}, + MRNUMBER = {763780}, +MRREVIEWER = {Michel Hilsum}, + DOI = {10.1016/0022-1236(84)90056-9}, + URL = {https://doi.org/10.1016/0022-1236(84)90056-9}, +} + +# Higson +@article {Higson88, + AUTHOR = {Higson, Nigel}, + TITLE = {Algebraic {K}-theory of stable {C}*-algebras}, + JOURNAL = {Adv. in Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {67}, + YEAR = {1988}, + NUMBER = {1}, + PAGES = {140}, + ISSN = {0001-8708}, + MRCLASS = {46L80 (19K14 46M20)}, + MRNUMBER = {922140}, +MRREVIEWER = {Cornel Pasnicu}, + DOI = {10.1016/0001-8708(88)90034-5}, + URL = {https://doi.org/10.1016/0001-8708(88)90034-5}, +} + +# Hirshberg-Rordam-Winter +@article {HirshbergRordamWinter07, + AUTHOR = {Hirshberg, Ilan and R{\o}rdam, Mikael and Winter, Wilhelm}, + TITLE = {$\mathcal{C}_0({X})$-algebras, stability and strongly self-absorbing {C}*-algebras}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {339}, + YEAR = {2007}, + NUMBER = {3}, + PAGES = {695--732}, + ISSN = {0025-5831}, + MRCLASS = {46L05 (47L40)}, + MRNUMBER = {2336064}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.1007/s00208-007-0129-8}, + URL = {https://doi.org/10.1007/s00208-007-0129-8}, +} + +# Hirshberg-Orovitz +@article {HirshbergOrovitz13, + AUTHOR = {Hirshberg, Ilan and Orovitz, Joav}, + TITLE = {Tracially $\mathcal{Z}$-absorbing {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {265}, + YEAR = {2013}, + NUMBER = {5}, + PAGES = {765--785}, + ISSN = {0022-1236}, + MRCLASS = {46L05}, + MRNUMBER = {3063095}, +MRREVIEWER = {Stuart A. White}, + DOI = {10.1016/j.jfa.2013.05.005}, + URL = {https://doi.org/10.1016/j.jfa.2013.05.005}, +} + +# Hirshberg-Winter +@article {HirshbergWinter07, + AUTHOR = {Hirshberg, Ilan and Winter, Wilhelm}, + TITLE = {{R}okhlin actions and self-absorbing {C}*-algebras}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {233}, + YEAR = {2007}, + NUMBER = {1}, + PAGES = {125--143}, + ISSN = {0030-8730}, + MRCLASS = {46L05 (46L55)}, + MRNUMBER = {2366371}, +MRREVIEWER = {Efren Ruiz}, + DOI = {10.2140/pjm.2007.233.125}, + URL = {https://doi.org/10.2140/pjm.2007.233.125}, +} + +@article {HirshbergWinter08, + AUTHOR = {Hirshberg, Ilan and Winter, Wilhelm}, + TITLE = {Permutations of strongly self-absorbing {C}*-algebras}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {19}, + YEAR = {2008}, + NUMBER = {9}, + PAGES = {1137--1145}, + ISSN = {0129-167X}, + MRCLASS = {46L05 (46L45 46L55)}, + MRNUMBER = {2458564}, +MRREVIEWER = {Yoshikazu Katayama}, + DOI = {10.1142/S0129167X08005011}, + URL = {https://doi.org/10.1142/S0129167X08005011}, +} + +# Hoschs, Kaad, Schemaitat +@article {HochsKaadSchemaitat18, + AUTHOR = {Hochs, Peter and Kaad, Jens and Schemaitat, Andr\'{e}}, + TITLE = {Algebraic {$K$}-theory and a semifinite {F}uglede-{K}adison + determinant}, + JOURNAL = {Ann. K-Theory}, + FJOURNAL = {Annals of K-Theory}, + VOLUME = {3}, + YEAR = {2018}, + NUMBER = {2}, + PAGES = {193--206}, + ISSN = {2379-1683,2379-1691}, + MRCLASS = {46L80}, + MRNUMBER = {3781426}, +MRREVIEWER = {Christopher\ Jack\ Bourne}, + DOI = {10.2140/akt.2018.3.193}, + URL = {https://doi.org/10.2140/akt.2018.3.193}, +} + +# Husemoller +@book {Husemoller66, + AUTHOR = {Husemoller, Dale}, + TITLE = {Fibre bundles}, + PUBLISHER = {McGraw-Hill Book Co., New York-London-Sydney}, + YEAR = {1966}, + PAGES = {xiv+300}, + MRCLASS = {57.30 (55.00)}, + MRNUMBER = {0229247}, +MRREVIEWER = {R. Williamson}, +} + +# Hurewicz-Wallman +@book {HurewiczWallman, + AUTHOR = {Hurewicz, Witold and Wallman, Henry}, + TITLE = {Dimension theory}, + SERIES = {Princeton Mathematical Series, vol. 4}, + PUBLISHER = {Princeton University Press, Princeton, N. J.}, + YEAR = {1941}, + PAGES = {vii+165}, + MRCLASS = {56.0X}, + MRNUMBER = {0006493}, +MRREVIEWER = {H. Whitney}, +} + + +### IIIIIIIIIIIIIIIIIIIIIIIIIIIIII + +# Ivanescu-Kucerovsky +@article{IvanescuKucerovsky23, + title={Villadsen algebras}, + author={Cristian Ivanescu and Dan Kucerovsky}, + year={2023}, + eprint={2306.03943}, + archivePrefix={arXiv}, + primaryClass={math.OA}, + NOTE={preprint}, +} + +# Izumi +@article{Izumi02, + title={Inclusions of simple {C}*-algebras}, + author={Izumi, Masaki}, + year={2002}, + volume = {547}, + pages = {97--138}, + journal = {J. Reine Angew. Math.}, + publisher={Walter de Gruyter GmbH \& Co. KG Berlin, Germany}, +} + +@article {Izumi04, + AUTHOR = {Izumi, Masaki}, + TITLE = {Finite group actions on {C}*-algebras with the {R}ohlin property. {I}}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {122}, + YEAR = {2004}, + NUMBER = {2}, + PAGES = {233--280}, + ISSN = {0012-7094}, + MRCLASS = {46L55 (19K35 46L35 46L40 46L80)}, + MRNUMBER = {2053753}, +MRREVIEWER = {Valentin Deaconu}, + DOI = {10.1215/S0012-7094-04-12221-3}, + URL = {https://doi.org/10.1215/S0012-7094-04-12221-3}, +} + +### JJJJJJJJJJJJJJJJJJJJJ +# Jiang +@article{Jiang97, + title={Nonstable {K}-theory for $\mathcal{Z}$-stable {C}*-algebras}, + author={Jiang, Xinhui}, + journal={arXiv preprint math/9707228}, + year={1997} +} + +# Jiang-Su +@article {JiangSu99, + AUTHOR = {Jiang, Xinhui and Su, Hongbing}, + TITLE = {On a simple unital projectionless {C}*-algebra}, + JOURNAL = {Amer. J. Math.}, + FJOURNAL = {American Journal of Mathematics}, + VOLUME = {121}, + YEAR = {1999}, + NUMBER = {2}, + PAGES = {359--413}, + ISSN = {0002-9327}, + MRCLASS = {46L35 (19K35 46L80)}, + MRNUMBER = {1680321}, +MRREVIEWER = {Vicumpriya S. Perera}, + URL = + {http://muse.jhu.edu/journals/american_journal_of_mathematics/v121/121.2jiang.pdf}, +} + +#Jolissaint +@article{Jolissaint16, + AUTHOR = {Jolissaint, Paul}, + TITLE = {Relative inner amenability and relative property gamma}, + JOURNAL = {Math. Scand.}, + FJOURNAL = {Mathematica Scandinavica}, + VOLUME = {119}, + YEAR = {2016}, + NUMBER = {2}, + PAGES = {293--319}, + ISSN = {0025-5521}, + MRCLASS = {22D25 (22D10 22F05 43A07 46L37)}, + MRNUMBER = {3570949}, +MRREVIEWER = {Jonathan M. Rosenberg}, + DOI = {10.7146/math.scand.a-24748}, + URL = {https://doi.org/10.7146/math.scand.a-24748}, +} + +@article {Jones80, + AUTHOR = {Jones, Vaughan F. R.}, + TITLE = {Actions of finite groups on the hyperfinite type {II}$_1$ factor}, + JOURNAL = {Mem. Amer. Math. Soc.}, + FJOURNAL = {Memoirs of the American Mathematical Society}, + VOLUME = {28}, + YEAR = {1980}, + NUMBER = {237}, + PAGES = {v+70}, + ISSN = {0065-9266,1947-6221}, + MRCLASS = {46L55 (20C99 46L40)}, + MRNUMBER = {587749}, +MRREVIEWER = {Marie\ Choda}, + DOI = {10.1090/memo/0237}, + URL = {https://doi.org/10.1090/memo/0237}, +} + +### KKKKKKKKKKKKKKKKKKKKKKKKK +# Kalantar-Kennedy +@article {KalantarKennedy17, + AUTHOR = {Kalantar, Mehrdad and Kennedy, Matthew}, + TITLE = {Boundaries of reduced {$C^*$}-algebras of discrete groups}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {727}, + YEAR = {2017}, + PAGES = {247--267}, + ISSN = {0075-4102,1435-5345}, + MRCLASS = {22D15 (20F67 22F05 46L10)}, + MRNUMBER = {3652252}, +MRREVIEWER = {Fernando\ Abadie}, + DOI = {10.1515/crelle-2014-0111}, + URL = {https://doi.org/10.1515/crelle-2014-0111}, +} + +# Kawahigashi-Sutherland-Takesaki +@article {KawahigashiSutherlandTakesaki92, + AUTHOR = {Kawahigashi, Y. and Sutherland, C. E. and Takesaki, M.}, + TITLE = {The structure of the automorphism group of an injective factor + and the cocycle conjugacy of discrete abelian group actions}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {169}, + YEAR = {1992}, + NUMBER = {1-2}, + PAGES = {105--130}, + ISSN = {0001-5962,1871-2509}, + MRCLASS = {46L40}, + MRNUMBER = {1179014}, +MRREVIEWER = {Yoshikazu\ Katayama}, + DOI = {10.1007/BF02392758}, + URL = {https://doi.org/10.1007/BF02392758}, +} + + +# Kawamuro +@article {Kawamuro99, + AUTHOR = {Kawamuro, Keiko}, + TITLE = {Central sequence subfactors and double commutant properties}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {10}, + YEAR = {1999}, + NUMBER = {1}, + PAGES = {53--77}, + ISSN = {0129-167X}, + MRCLASS = {46L37}, + MRNUMBER = {1678538}, +MRREVIEWER = {Carl Winsl{\o}w}, + DOI = {10.1142/S0129167X99000033}, + URL = {https://doi.org/10.1142/S0129167X99000033}, +} + +# Kennedy-Schafhauser +@article {KennedySchafhauser19, + AUTHOR = {Kennedy, Matthew and Schafhauser, Christopher}, + TITLE = {Noncommutative boundaries and the ideal structure of reduced + crossed products}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {168}, + YEAR = {2019}, + NUMBER = {17}, + PAGES = {3215--3260}, + ISSN = {0012-7094,1547-7398}, + MRCLASS = {46L35 (43A65 47L65)}, + MRNUMBER = {4030364}, + DOI = {10.1215/00127094-2019-0032}, + URL = {https://doi.org/10.1215/00127094-2019-0032}, +} + +# Kennedy-Shamovich +@article {KennedyShamovich22, + AUTHOR = {Kennedy, Matthew and Shamovich, Eli}, + TITLE = {Noncommutative {C}hoquet simplices}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {382}, + YEAR = {2022}, + NUMBER = {3-4}, + PAGES = {1591--1629}, + ISSN = {0025-5831,1432-1807}, + MRCLASS = {46A55 (46L07 46L52 47A20 47L25)}, + MRNUMBER = {4403230}, +MRREVIEWER = {Andrey\ I.\ Zahariev}, + DOI = {10.1007/s00208-021-02261-z}, + URL = {https://doi.org/10.1007/s00208-021-02261-z}, +} + +# Kirchberg +@inproceedings {Kirchberg95, + AUTHOR = {Kirchberg, Eberhard}, + TITLE = {Exact {${\rm C}^*$}-algebras, tensor products, and the + classification of purely infinite algebras}, + BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of + {M}athematicians, {V}ol. 1, 2 ({Z}\"{u}rich, 1994)}, + PAGES = {943--954}, + PUBLISHER = {Birkh\"{a}user, Basel}, + YEAR = {1995}, + ISBN = {3-7643-5153-5}, + MRCLASS = {46L05 (46L35 46M05 46M15)}, + MRNUMBER = {1403994}, +MRREVIEWER = {Robert\ S.\ Doran}, +} + +@incollection {Kirchberg06, + AUTHOR = {Kirchberg, Eberhard}, + TITLE = {Central sequences in {C}*-algebras and strongly purely + infinite algebras}, + BOOKTITLE = {Operator {A}lgebras: {T}he {A}bel {S}ymposium 2004}, + SERIES = {Abel Symp.}, + VOLUME = {1}, + PAGES = {175--231}, + PUBLISHER = {Springer, Berlin}, + YEAR = {2006}, + MRCLASS = {46L05}, + MRNUMBER = {2265050}, +MRREVIEWER = {Wilhelm Winter}, + DOI = {10.1007/978-3-540-34197-0\_10}, + URL = {https://doi.org/10.1007/978-3-540-34197-0_10}, +} + +# Kirchberg-Phillips +@article {KirchbergPhillips00, + AUTHOR = {Kirchberg, Eberhard and Phillips, N. Christopher}, + TITLE = {Embedding of exact {C}*-algebras in the {C}untz algebra $\mathcal{O}_2$}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {525}, + YEAR = {2000}, + PAGES = {17--53}, + ISSN = {0075-4102}, + MRCLASS = {46L05 (19K56 46L35 46L80)}, + MRNUMBER = {1780426}, +MRREVIEWER = {Mikael R{\o}rdam}, + DOI = {10.1515/crll.2000.065}, + URL = {https://doi.org/10.1515/crll.2000.065}, +} + +# Kirchberg-Wassermann +@article {KirchbergWassermann99a, + AUTHOR = {Kirchberg, Eberhard and Wassermann, Simon}, + TITLE = {Exact groups and continuous bundles of {C}*-algebras}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {315}, + YEAR = {1999}, + NUMBER = {2}, + PAGES = {169--203}, + ISSN = {0025-5831,1432-1807}, + MRCLASS = {46L05 (22D05 46M20)}, + MRNUMBER = {1721796}, +MRREVIEWER = {Michael\ Frank}, + DOI = {10.1007/s002080050364}, + URL = {https://doi.org/10.1007/s002080050364}, +} + +@article {KirchbergWassermann99b, + AUTHOR = {Kirchberg, Eberhard and Wassermann, Simon}, + TITLE = {Permanence properties of {C}*-exact groups}, + JOURNAL = {Doc. Math.}, + FJOURNAL = {Documenta Mathematica}, + VOLUME = {4}, + YEAR = {1999}, + PAGES = {513--558}, + ISSN = {1431-0635,1431-0643}, + MRCLASS = {46L05 (22D05 46L80)}, + MRNUMBER = {1725812}, +MRREVIEWER = {Erik\ B\'{e}dos}, +} + + +# Kishimoto +@article {Kishimoto00, + AUTHOR = {Kishimoto, A.}, + TITLE = {{R}ohlin property for shift automorphisms}, + JOURNAL = {Rev. Math. Phys.}, + FJOURNAL = {Reviews in Mathematical Physics. A Journal for Both Review and + Original Research Papers in the Field of Mathematical Physics}, + VOLUME = {12}, + YEAR = {2000}, + NUMBER = {7}, + PAGES = {965--980}, + ISSN = {0129-055X}, + MRCLASS = {46L40 (46L05 46L55)}, + MRNUMBER = {1782691}, +MRREVIEWER = {Claire Anantharaman-Delaroche}, + DOI = {10.1142/S0129055X00000368}, + URL = {https://doi.org/10.1142/S0129055X00000368}, +} + +# Kumjian +@article {Kumjian88, + AUTHOR = {Kumjian, Alexander}, + TITLE = {An involutive automorphism of the {B}unce-{D}eddens algebra}, + JOURNAL = {C. R. Math. Rep. Acad. Sci. Canada}, + FJOURNAL = {La Soci\'{e}t\'{e} Royale du Canada. L'Academie des Sciences. Comptes + Rendus Math\'{e}matiques. (Mathematical Reports)}, + VOLUME = {10}, + YEAR = {1988}, + NUMBER = {5}, + PAGES = {217--218}, + ISSN = {0706-1994}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {962104}, +MRREVIEWER = {J. W. Bunce}, +} + +### LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL +# Lee-Osaka +@article{LeeOsaka23, + title={On permanence of regularity properties}, + author={Lee, Hyun Ho and Osaka, Hiroyuki}, + journal={Journal of Topology and Analysis}, + year={2023}, + publisher={World Scientific} +} + +# Luck-Rordam +@article {LuckRordam93, + AUTHOR = {L\"{u}ck, Wolfgang and R{\o}rdam, Mikael}, + TITLE = {Algebraic {K}-theory of von {N}eumann algebras}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {7}, + YEAR = {1993}, + NUMBER = {6}, + PAGES = {517--536}, + ISSN = {0920-3036}, + MRCLASS = {46L80 (19B28 19J10 19K56 19K99)}, + MRNUMBER = {1268591}, +MRREVIEWER = {Jonathan M. Rosenberg}, + DOI = {10.1007/BF00961216}, + URL = {https://doi.org/10.1007/BF00961216}, +} + +### MMMMMMMMMMMMMMMMMMMMMMM +# Maclane +@book {Maclane12, + AUTHOR = {MacLane, Saunders}, + TITLE = {Homology}, + SERIES = {Die Grundlehren der mathematischen Wissenschaften, Band 114}, + EDITION = {first}, + PUBLISHER = {Springer-Verlag, Berlin-New York}, + YEAR = {1967}, + PAGES = {x+422}, + MRCLASS = {18-02}, + MRNUMBER = {0349792}, +} + +#Manor +@article {Manor21, + AUTHOR = {Manor, Nicholas}, + TITLE = {Exactness versus {C}*-exactness for certain + non-discrete groups}, + JOURNAL = {Integral Equations Operator Theory}, + FJOURNAL = {Integral Equations and Operator Theory}, + VOLUME = {93}, + YEAR = {2021}, + NUMBER = {3}, + PAGES = {Paper No. 20, 13}, + ISSN = {0378-620X,1420-8989}, + MRCLASS = {22D25 (46L06)}, + MRNUMBER = {4249041}, +MRREVIEWER = {Vladimir\ Manuilov}, + DOI = {10.1007/s00020-021-02634-8}, + URL = {https://doi.org/10.1007/s00020-021-02634-8}, +} + +# Marcoux-Popov +@article {MarcouxPopov16, + AUTHOR = {Marcoux, Laurent W. and Popov, Alexey I.}, + TITLE = {Abelian, amenable operator algebras are similar to + {C}*-algebras}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {165}, + YEAR = {2016}, + NUMBER = {12}, + PAGES = {2391--2406}, + ISSN = {0012-7094,1547-7398}, + MRCLASS = {46J05 (47L10 47L30)}, + MRNUMBER = {3544284}, +MRREVIEWER = {Damon\ Martin\ Hay}, + DOI = {10.1215/00127094-3619791}, + URL = {https://doi.org/10.1215/00127094-3619791}, +} + + +# Matui-Sato +@article {MatuiSato12, + AUTHOR = {Matui, Hiroki and Sato, Yasuhiko}, + TITLE = {Strict comparison and $\mathcal{Z}$-absorption of nuclear {C}*-algebras}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {209}, + YEAR = {2012}, + NUMBER = {1}, + PAGES = {179--196}, + ISSN = {0001-5962}, + MRCLASS = {46L05}, + MRNUMBER = {2979512}, +MRREVIEWER = {Caleb Eckhardt}, + DOI = {10.1007/s11511-012-0084-4}, + URL = {https://doi.org/10.1007/s11511-012-0084-4}, +} + +#McDuff +@article {McDuff69, + AUTHOR = {McDuff, Dusa}, + TITLE = {Uncountably many {II}$_1$ factors}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {90}, + YEAR = {1969}, + PAGES = {372--377}, + ISSN = {0003-486X}, + MRCLASS = {46.65}, + MRNUMBER = {259625}, +MRREVIEWER = {Z. Takeda}, + DOI = {10.2307/1970730}, + URL = {https://doi.org/10.2307/1970730}, +} + +@article {McDuff70, + AUTHOR = {McDuff, Dusa}, + TITLE = {Central sequences and the hyperfinite factor}, + JOURNAL = {Proc. London Math. Soc. (3)}, + FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, + VOLUME = {21}, + YEAR = {1970}, + PAGES = {443--461}, + ISSN = {0024-6115}, + MRCLASS = {46.65}, + MRNUMBER = {281018}, +MRREVIEWER = {M. Takesaki}, + DOI = {10.1112/plms/s3-21.3.443}, + URL = {https://doi.org/10.1112/plms/s3-21.3.443}, +} + +# Munkres +@book {Munkres, + AUTHOR = {Munkres, James R.}, + TITLE = {Topology}, + NOTE = {Second edition of [ MR0464128]}, + PUBLISHER = {Prentice Hall, Inc., Upper Saddle River, NJ}, + YEAR = {2000}, + PAGES = {xvi+537}, + ISBN = {0-13-181629-2}, + MRCLASS = {54-01}, + MRNUMBER = {3728284}, +} + +# Murphy +@book {Murphybook, + AUTHOR = {Murphy, Gerard J.}, + TITLE = {{C}*-algebras and operator theory}, + PUBLISHER = {Academic Press, Inc., Boston, MA}, + YEAR = {1990}, + PAGES = {x+286}, + ISBN = {0-12-511360-9}, + MRCLASS = {46Lxx (46-01)}, + MRNUMBER = {1074574}, +MRREVIEWER = {E. Gerlach}, +} + +# Murray-von Neumann +@article {MvNI, + AUTHOR = {Murray, Francis J. and von Neumann, John}, + TITLE = {On rings of operators}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {37}, + YEAR = {1936}, + NUMBER = {1}, + PAGES = {116--229}, + ISSN = {0003-486X,1939-8980}, + MRCLASS = {99-04}, + MRNUMBER = {1503275}, + DOI = {10.2307/1968693}, + URL = {https://doi.org/10.2307/1968693}, +} + +@article {MvNIV, + AUTHOR = {Murray, Francis J. and von Neumann, John}, + TITLE = {On rings of operators. {IV}}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {44}, + YEAR = {1943}, + PAGES = {716--808}, + ISSN = {0003-486X}, + MRCLASS = {46.0X}, + MRNUMBER = {9096}, +MRREVIEWER = {E. R. Lorch}, + DOI = {10.2307/1969107}, + URL = {https://doi.org/10.2307/1969107}, +} + +# Mygind +@article{Mygind01, + title={Classification of certain simple {C}*-algebras with torsion in {K}1}, + author={Mygind, Jesper}, + journal={Canadian Journal of Mathematics}, + volume={53}, + number={6}, + pages={1223--1308}, + year={2001}, + publisher={Cambridge University Press} +} +@article {MR1863849, + AUTHOR = {Mygind, Jesper}, + TITLE = {Classification of certain simple {C}*-algebras with torsion in {$K_1$}}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {53}, + YEAR = {2001}, + NUMBER = {6}, + PAGES = {1223--1308}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (19K14 46L80)}, + MRNUMBER = {1863849}, +MRREVIEWER = {Xiao Chun Fang}, + DOI = {10.4153/CJM-2001-046-2}, + URL = {https://doi.org/10.4153/CJM-2001-046-2}, +} + +### NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +# Ng +@article {Ng06, + AUTHOR = {Ng, Ping W.}, + TITLE = {Amenability of the sequence of unitary groups associated with + a {$C^*$}-algebra}, + JOURNAL = {Indiana Univ. Math. J.}, + FJOURNAL = {Indiana University Mathematics Journal}, + VOLUME = {55}, + YEAR = {2006}, + NUMBER = {4}, + PAGES = {1389--1400}, + ISSN = {0022-2518,1943-5258}, + MRCLASS = {46L05 (43A07)}, + MRNUMBER = {2269417}, +MRREVIEWER = {A.\ G.\ Myasnikov}, + DOI = {10.1512/iumj.2006.55.2791}, + URL = {https://doi.org/10.1512/iumj.2006.55.2791}, +} + +@article {Ng14, + AUTHOR = {Ng, Ping W.}, + TITLE = {The kernel of the determinant map on certain simple {C}*-algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {71}, + YEAR = {2014}, + NUMBER = {2}, + PAGES = {341--379}, + ISSN = {0379-4024}, + MRCLASS = {46L05 (46L80 47B47 47C15)}, + MRNUMBER = {3214642}, +MRREVIEWER = {Takahiro Sudo}, + DOI = {10.7900/jot.2012apr01.1953}, + URL = {https://doi.org/10.7900/jot.2012apr01.1953}, +} + +# Ng-Robert +@article {NgRobert16, + AUTHOR = {Ng, Ping W. and Robert, Leonel}, + TITLE = {Sums of commutators in pure {C}*-algebras}, + JOURNAL = {M\"{u}nster J. Math.}, + FJOURNAL = {M\"{u}nszter Journal of Mathematics}, + VOLUME = {9}, + YEAR = {2016}, + NUMBER = {1}, + PAGES = {121--154}, + ISSN = {1867-5778,1867-5786}, + MRCLASS = {46L05 (46L35 46L57 47B47)}, + MRNUMBER = {3549546}, +MRREVIEWER = {Bojan\ P.\ Magajna}, + DOI = {10.17879/35209721075}, + URL = {https://doi.org/10.17879/35209721075}, +} + +@article {NgRobert17, + AUTHOR = {Ng, Ping W. and Robert, Leonel}, + TITLE = {The kernel of the determinant map on pure {C}*-algebras}, + JOURNAL = {Houston J. Math.}, + FJOURNAL = {Houston Journal of Mathematics}, + VOLUME = {43}, + YEAR = {2017}, + NUMBER = {1}, + PAGES = {139--168}, + ISSN = {0362-1588}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {3647937}, +MRREVIEWER = {Daniele Puglisi}, + DOI = {10.1007/s11139-016-9879-9}, + URL = {https://doi.org/10.1007/s11139-016-9879-9}, +} + +# Nielsen-Thomsen +@article {NielsenThomsen96, + AUTHOR = {Nielsen, Karen E. and Thomsen, Klaus}, + TITLE = {Limits of circle algebras}, + JOURNAL = {Exposition. Math.}, + FJOURNAL = {Expositiones Mathematicae. International Journal}, + VOLUME = {14}, + YEAR = {1996}, + NUMBER = {1}, + PAGES = {17--56}, + ISSN = {0723-0869}, + MRCLASS = {46L80 (19K14 46L05)}, + MRNUMBER = {1382013}, +MRREVIEWER = {Sze-Kai Tsui}, +} + +### OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO + +@book {Ocneanu85, + AUTHOR = {Ocneanu, Adrian}, + TITLE = {Actions of discrete amenable groups on von {N}eumann algebras}, + SERIES = {Lecture Notes in Mathematics}, + VOLUME = {1138}, + PUBLISHER = {Springer-Verlag, Berlin}, + YEAR = {1985}, + PAGES = {iv+115}, + ISBN = {3-540-15663-1}, + MRCLASS = {46L55 (46L35 46L40)}, + MRNUMBER = {807949}, +MRREVIEWER = {Hisashi\ Choda}, + DOI = {10.1007/BFb0098579}, + URL = {https://doi.org/10.1007/BFb0098579}, +} + +@article {OsakaTeruya18, + AUTHOR = {Osaka, Hiroyuki and Teruya, Tamotsu}, + TITLE = {The {J}iang--{S}u absorption for inclusions of unital {C}*-algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {70}, + YEAR = {2018}, + NUMBER = {2}, + PAGES = {400--425}, + ISSN = {0008-414X}, + MRCLASS = {46L55 (46L35)}, + MRNUMBER = {3759005}, +MRREVIEWER = {Daniele Puglisi}, + DOI = {10.4153/CJM-2017-033-7}, + URL = {https://doi.org/10.4153/CJM-2017-033-7}, +} + +# Ozawa +@article {Ozawa13, + AUTHOR = {Ozawa, Narutaka}, + TITLE = {{D}ixmier approximation and symmetric amenability for {C}*-algebras}, + JOURNAL = {J. Math. Sci. Univ. Tokyo}, + FJOURNAL = {The University of Tokyo. Journal of Mathematical Sciences}, + VOLUME = {20}, + YEAR = {2013}, + NUMBER = {3}, + PAGES = {349--374}, + ISSN = {1340-5705}, + MRCLASS = {46L05 (46L10)}, + MRNUMBER = {3156986}, +MRREVIEWER = {William Paschke}, +} + +@article{Ozawa23, + title={Amenability for unitary groups of simple monotracial {C}*-algebras}, + author={Ozawa, Narutaka}, + journal={arXiv:2307.08267}, + year={2023} +} + +### PPPPPPPPPPPPPPPPPPPPPPP +# Paterson +@article{Paterson83, + AUTHOR = {Paterson, Alan L. T.}, + TITLE = {Harmonic analysis on unitary groups}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {53}, + YEAR = {1983}, + NUMBER = {3}, + PAGES = {203--223}, + ISSN = {0022-1236}, + MRCLASS = {22D10 (22D25 43A65 46L05)}, + MRNUMBER = {724026}, +MRREVIEWER = {J. Gil de Lamadrid}, + DOI = {10.1016/0022-1236(83)90031-9}, + URL = {https://doi.org/10.1016/0022-1236(83)90031-9}, +} + +@article {Paterson92, + AUTHOR = {Paterson, Alan L. T.}, + TITLE = {Nuclear {C}*-algebras have amenable unitary groups}, + JOURNAL = {Proc. Amer. Math. Soc.}, + FJOURNAL = {Proceedings of the American Mathematical Society}, + VOLUME = {114}, + YEAR = {1992}, + NUMBER = {3}, + PAGES = {719--721}, + ISSN = {0002-9939}, + MRCLASS = {46L05}, + MRNUMBER = {1076577}, +MRREVIEWER = {C. J. K. Batty}, + DOI = {10.2307/2159395}, + URL = {https://doi.org/10.2307/2159395}, +} + +# Paulsen +@book{Paulsenbook, + AUTHOR = {Paulsen, Vern}, + TITLE = {Completely bounded maps and operator algebras}, + SERIES = {Cambridge Studies in Advanced Mathematics}, + VOLUME = {78}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {2002}, + PAGES = {xii+300}, + ISBN = {0-521-81669-6}, + MRCLASS = {46L07 (47A20 47L30)}, + MRNUMBER = {1976867}, +MRREVIEWER = {Christian Le Merdy}, +} + +# Phillips +@article {Phillips92, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {The rectifiable metric on the space of projections in a {C}*-algebra}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {3}, + YEAR = {1992}, + NUMBER = {5}, + PAGES = {679--698}, + ISSN = {0129-167X}, + MRCLASS = {46L05}, + MRNUMBER = {1189681}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1142/S0129167X92000333}, + URL = {https://doi.org/10.1142/S0129167X92000333}, +} + +@article {Phillips95, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {Exponential length and traces}, + JOURNAL = {Proc. Roy. Soc. Edinburgh Sect. A}, + FJOURNAL = {Proceedings of the Royal Society of Edinburgh. Section A. + Mathematics}, + VOLUME = {125}, + YEAR = {1995}, + NUMBER = {1}, + PAGES = {13--29}, + ISSN = {0308-2105}, + MRCLASS = {46L05}, + MRNUMBER = {1318621}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1017/S0308210500030730}, + URL = {https://doi.org/10.1017/S0308210500030730}, +} + +@article {Phillips00, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {A classification theorem for nuclear purely infinite simple {C}*-algebras}, + JOURNAL = {Doc. Math.}, + FJOURNAL = {Documenta Mathematica}, + VOLUME = {5}, + YEAR = {2000}, + PAGES = {49--114}, + ISSN = {1431-0635}, + MRCLASS = {46L05 (19K56 46L35 46L80)}, + MRNUMBER = {1745197}, +MRREVIEWER = {Mikael R{\o}rdam}, +} + +@article {Phillips01, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {Continuous-trace {C}*-algebras not isomorphic to their opposite algebras}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {12}, + YEAR = {2001}, + NUMBER = {3}, + PAGES = {263--275}, + ISSN = {0129-167X}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {1841515}, +MRREVIEWER = {Takahiro Sudo}, + DOI = {10.1142/S0129167X01000642}, + URL = {https://doi.org/10.1142/S0129167X01000642}, +} + +@article{Phillips04, + title={A simple separable {C}*-algebra not isomorphic to its opposite algebra}, + author={Phillips, N. Christopher}, + journal={Proceedings of the American Mathematical Society}, + volume={132}, + number={10}, + pages={2997--3005}, + year={2004} +} +@article {Phillips04, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {A simple separable {C}*-algebra not isomorphic to its opposite algebra}, + JOURNAL = {Proc. Amer. Math. Soc.}, + FJOURNAL = {Proceedings of the American Mathematical Society}, + VOLUME = {132}, + YEAR = {2004}, + NUMBER = {10}, + PAGES = {2997--3005}, + ISSN = {0002-9939}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {2063121}, +MRREVIEWER = {Hiroyuki Osaka}, + DOI = {10.1090/S0002-9939-04-07330-7}, + URL = {https://doi.org/10.1090/S0002-9939-04-07330-7}, +} + +@article{Phillips12, + title={The tracial {R}okhlin property is generic}, + author={Phillips, N. Christopher}, + journal={arXiv:1209.3859}, + year={2012} +} + +# Phillips-Viola +@article {PhillipsViola13, + AUTHOR = {Phillips, N. Christopher and Viola, Maria Grazia}, + TITLE = {A simple separable exact {C}*-algebra not anti-isomorphic to itself}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {355}, + YEAR = {2013}, + NUMBER = {2}, + PAGES = {783--799}, + ISSN = {0025-5831}, + MRCLASS = {46L35 (46L09 46L37 46L40)}, + MRNUMBER = {3010147}, +MRREVIEWER = {Snigdhayan Mahanta}, + DOI = {10.1007/s00208-011-0755-z}, + URL = {https://doi.org/10.1007/s00208-011-0755-z}, +} + +# Pimsner-Voiculescu +@article {PimsnerVoiculescu80, + AUTHOR = {Pimsner, Mihai and Voiculescu, Dan}, + TITLE = {Imbedding the irrational rotation {C}*-algebra into {AF}-algebra}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {4}, + YEAR = {1980}, + NUMBER = {2}, + PAGES = {201--210}, + ISSN = {0379-4024}, + MRCLASS = {46L35 (16A54)}, + MRNUMBER = {595412}, +MRREVIEWER = {David Handelman}, +} + +# Pitts +@article {Pitts17, + AUTHOR = {Pitts, David R.}, + TITLE = {Structure for regular inclusions. {I}}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {78}, + YEAR = {2017}, + NUMBER = {2}, + PAGES = {357--416}, + ISSN = {0379-4024,1841-7744}, + MRCLASS = {46L05 (47L30)}, + MRNUMBER = {3725511}, +MRREVIEWER = {Michael\ S.\ Anoussis}, + DOI = {10.7900/jot}, + URL = {https://doi.org/10.7900/jot}, +} + +@article {Pitts21StructureII, + AUTHOR = {Pitts, David R.}, + TITLE = {Structure for regular inclusions. {II}: {C}artan envelopes, + pseudo-expectations and twists}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {281}, + YEAR = {2021}, + NUMBER = {1}, + PAGES = {Paper No. 108993, 66}, + ISSN = {0022-1236,1096-0783}, + MRCLASS = {46L05 (22A22 46L07)}, + MRNUMBER = {4234857}, +MRREVIEWER = {Preeti\ Luthra}, + DOI = {10.1016/j.jfa.2021.108993}, + URL = {https://doi.org/10.1016/j.jfa.2021.108993}, +} + +@article{Pitts21Normalizers, + title={Normalizers and approximate units for inclusions of C*-algebras}, + author={Pitts, David R}, + journal={arXiv:2109.00856}, + year={2021} +} + +# Pitts-Smith-Zarikian +@article{PittsSmithZarikian23, + title={Norming in Discrete Crossed Products}, + author={Pitts, David R and Smith, Roger R and Zarikian, Vrej}, + journal={arXiv:2307.15571}, + year={2023} +} + +# Pitts-Zarikian +@article {PittsZarikian15, + AUTHOR = {Pitts, David R. and Zarikian, Vrej}, + TITLE = {Unique pseudo-expectations for {C}*-inclusions}, + JOURNAL = {Illinois J. Math.}, + FJOURNAL = {Illinois Journal of Mathematics}, + VOLUME = {59}, + YEAR = {2015}, + NUMBER = {2}, + PAGES = {449--483}, + ISSN = {0019-2082,1945-6581}, + MRCLASS = {46L05 (46L07 46L10 46M10)}, + MRNUMBER = {3499520}, +MRREVIEWER = {Bernard\ Russo}, + URL = {http://projecteuclid.org/euclid.ijm/1462450709}, +} + +# Popa +@article {Popa83, + AUTHOR = {Popa, Sorin}, + TITLE = {Orthogonal pairs of *-subalgebras in finite von {N}eumann algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {9}, + YEAR = {1983}, + NUMBER = {2}, + PAGES = {253--268}, + ISSN = {0379-4024}, + MRCLASS = {46L10 (16A30 20C07)}, + MRNUMBER = {703810}, +MRREVIEWER = {G. A. Elliott}, +} + + +@article {Popa89, + AUTHOR = {Popa, Sorin}, + TITLE = {Sousfacteurs, actions des groupes et cohomologie}, + JOURNAL = {C. R. Acad. Sci. Paris S\'{e}r. I Math.}, + FJOURNAL = {Comptes Rendus de l'Acad\'{e}mie des Sciences. S\'{e}rie I. + Math\'{e}matique}, + VOLUME = {309}, + YEAR = {1989}, + NUMBER = {12}, + PAGES = {771--776}, + ISSN = {0764-4442}, + MRCLASS = {46L37 (22D25 22D35 46L55 46L80)}, + MRNUMBER = {1054961}, +MRREVIEWER = {Masatoshi Enomoto}, +} + +@article {Popa00, + AUTHOR = {Popa, Sorin}, + TITLE = {On the relative {D}ixmier property for inclusions of {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {171}, + YEAR = {2000}, + NUMBER = {1}, + PAGES = {139--154}, + ISSN = {0022-1236}, + MRCLASS = {46L37 (46L05)}, + MRNUMBER = {1742862}, +MRREVIEWER = {H. Halpern}, + DOI = {10.1006/jfan.1999.3536}, + URL = {https://doi.org/10.1006/jfan.1999.3536}, +} + +### RRRRRRRRRRRRRRRRRRRRRRRRRR +# Raeburn +@book {RaeburnBook, + AUTHOR = {Raeburn, Iain}, + TITLE = {Graph algebras}, + SERIES = {CBMS Regional Conference Series in Mathematics}, + VOLUME = {103}, + PUBLISHER = {Published for the Conference Board of the Mathematical + Sciences, Washington, DC; by the American Mathematical + Society, Providence, RI}, + YEAR = {2005}, + PAGES = {vi+113}, + ISBN = {0-8218-3660-9}, + MRCLASS = {46L05 (22D25)}, + MRNUMBER = {2135030}, +MRREVIEWER = {Mark Tomforde}, + DOI = {10.1090/cbms/103}, + URL = {https://doi.org/10.1090/cbms/103}, +} + +#Rieffel +@article {Rieffel87, + AUTHOR = {Rieffel, Marc A.}, + TITLE = {The homotopy groups of the unitary groups of noncommutative tori}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {17}, + YEAR = {1987}, + NUMBER = {2}, + PAGES = {237--254}, + ISSN = {0379-4024}, + MRCLASS = {22D25 (46L55 46L80)}, + MRNUMBER = {887221}, +MRREVIEWER = {Gustavo Corach}, +} + +# Robert +@article {Robert12, + AUTHOR = {Robert, Leonel}, + TITLE = {Classification of inductive limits of 1-dimensional {NCCW} + complexes}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {231}, + YEAR = {2012}, + NUMBER = {5}, + PAGES = {2802--2836}, + ISSN = {0001-8708}, + MRCLASS = {46L35}, + MRNUMBER = {2970466}, +MRREVIEWER = {Changguo Wei}, + DOI = {10.1016/j.aim.2012.07.010}, + URL = {https://doi.org/10.1016/j.aim.2012.07.010}, +} + +# Rordam +@article {Rordam91, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {On the structure of simple {C}*-algebras tensored with a {UHF}-algebra}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {100}, + YEAR = {1991}, + NUMBER = {1}, + PAGES = {1--17}, + ISSN = {0022-1236}, + MRCLASS = {46L05}, + MRNUMBER = {1124289}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.1016/0022-1236(91)90098-P}, + URL = {https://doi.org/10.1016/0022-1236(91)90098-P}, +} + + +@article {Rordam92, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {On the structure of simple {C}*-algebras tensored with a {UHF}-algebra. {II}}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {107}, + YEAR = {1992}, + NUMBER = {2}, + PAGES = {255--269}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L85)}, + MRNUMBER = {1172023}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.1016/0022-1236(92)90106-S}, + URL = {https://doi.org/10.1016/0022-1236(92)90106-S}, +} + +@article {Rordam93, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Classification of inductive limits of {C}untz algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {440}, + YEAR = {1993}, + PAGES = {175--200}, + ISSN = {0075-4102}, + MRCLASS = {46L05 (19K99 46L80)}, + MRNUMBER = {1225963}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1515/crll.1993.440.175}, + URL = {https://doi.org/10.1515/crll.1993.440.175}, +} + +@article {Rordam95, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Classification of certain infinite simple {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {131}, + YEAR = {1995}, + NUMBER = {2}, + PAGES = {415--458}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (19K35 46L35 46L80)}, + MRNUMBER = {1345038}, +MRREVIEWER = {Judith A. Packer}, + DOI = {10.1006/jfan.1995.1095}, + URL = {https://doi.org/10.1006/jfan.1995.1095}, +} + +@incollection {RordamBook, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Classification of nuclear, simple {C}*-algebras}, + BOOKTITLE = {Classification of nuclear {C}*-algebras. {E}ntropy in + operator algebras}, + SERIES = {Encyclopaedia Math. Sci.}, + VOLUME = {126}, + PAGES = {1--145}, + PUBLISHER = {Springer, Berlin}, + YEAR = {2002}, + MRCLASS = {46L05 (19K56 46L35 46L80)}, + MRNUMBER = {1878882}, +MRREVIEWER = {Judith A. Packer}, + DOI = {10.1007/978-3-662-04825-2\_1}, + URL = {https://doi.org/10.1007/978-3-662-04825-2_1}, +} + +@book {RordamKBook, + AUTHOR = {R{\o}rdam, Mikael and Larsen, Flemming and Laustsen, Niels J.}, + TITLE = {An introduction to {K}-theory for {C}*-algebras}, + SERIES = {London Mathematical Society Student Texts}, + VOLUME = {49}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {2000}, + PAGES = {xii+242}, + ISBN = {0-521-78334-8; 0-521-78944-3}, + MRCLASS = {46-01 (19K35 46L80)}, + MRNUMBER = {1783408}, +MRREVIEWER = {\'{E}ric Leichtnam}, +} + +@article {Rordam03, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {A simple {C}*-algebra with a finite and an infinite projection}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {191}, + YEAR = {2003}, + NUMBER = {1}, + PAGES = {109--142}, + ISSN = {0001-5962}, + MRCLASS = {46L05}, + MRNUMBER = {2020420}, +MRREVIEWER = {Berndt Brenken}, + DOI = {10.1007/BF02392697}, + URL = {https://doi.org/10.1007/BF02392697}, +} + +@article {Rordam04, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {The stable and the real rank of $\mathcal{Z}$-absorbing {C}*-algebras}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {15}, + YEAR = {2004}, + NUMBER = {10}, + PAGES = {1065--1084}, + ISSN = {0129-167X}, + MRCLASS = {46L35 (19K14 46L05 46L06)}, + MRNUMBER = {2106263}, +MRREVIEWER = {Nadia S. Larsen}, + DOI = {10.1142/S0129167X04002661}, + URL = {https://doi.org/10.1142/S0129167X04002661}, +} + +@article {Rordam23, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Irreducible inclusions of simple {C}*-algebras}, + JOURNAL = {Enseign. Math.}, + FJOURNAL = {L'Enseignement Math\'{e}matique}, + VOLUME = {69}, + YEAR = {2023}, + NUMBER = {3-4}, + PAGES = {275--314}, + ISSN = {0013-8584,2309-4672}, + MRCLASS = {46L05 (46L35 46L55)}, + MRNUMBER = {4599249}, + DOI = {10.4171/lem/1051}, + URL = {https://doi.org/10.4171/lem/1051}, +} + +# Rordam-Winter +@article {RordamWinter10, + AUTHOR = {R{\o}rdam, Mikael and Winter, Wilhelm}, + TITLE = {The {J}iang-{S}u algebra revisited}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {642}, + YEAR = {2010}, + PAGES = {129--155}, + ISSN = {0075-4102}, + MRCLASS = {46L35 (46L05 46L85)}, + MRNUMBER = {2658184}, +MRREVIEWER = {Hiroyuki Osaka}, + DOI = {10.1515/CRELLE.2010.039}, + URL = {https://doi.org/10.1515/CRELLE.2010.039}, +} + +# Rosenberg +@article {Rosenberg89, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {Continuous-trace algebras from the bundle theoretic point of + view}, + JOURNAL = {J. Austral. Math. Soc. Ser. A}, + FJOURNAL = {Australian Mathematical Society. Journal. Series A. Pure + Mathematics and Statistics}, + VOLUME = {47}, + YEAR = {1989}, + NUMBER = {3}, + PAGES = {368--381}, + ISSN = {0263-6115}, + MRCLASS = {46L80 (19K99 46L10 46M20 55R10)}, + MRNUMBER = {1018964}, +MRREVIEWER = {Claude Schochet}, +} + +@incollection {Rosenberg04, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {Comparison between algebraic and topological {K}-theory for + {B}anach algebras and {C}*-algebras}, + BOOKTITLE = {Handbook of {$K$}-theory. {V}ol. 1, 2}, + PAGES = {843--874}, + PUBLISHER = {Springer, Berlin}, + YEAR = {2005}, + MRCLASS = {46L80 (19K99 46H99)}, + MRNUMBER = {2181834}, +MRREVIEWER = {Efton Park}, + DOI = {10.1007/978-3-540-27855-9\_16}, + URL = {https://doi.org/10.1007/978-3-540-27855-9_16}, +} + + +@article {Rosenberg97, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {The algebraic {$K$}-theory of operator algebras}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {12}, + YEAR = {1997}, + NUMBER = {1}, + PAGES = {75--99}, + ISSN = {0920-3036}, + MRCLASS = {19K99 (19D35 19D50 19L41 46L80)}, + MRNUMBER = {1466624}, +MRREVIEWER = {Hiroshi Takai}, + DOI = {10.1023/A:1007736420938}, + URL = {https://doi.org/10.1023/A:1007736420938}, +} + +@book {Rosenberg94, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {Algebraic {$K$}-theory and its applications}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {147}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1994}, + PAGES = {x+392}, + ISBN = {0-387-94248-3}, + MRCLASS = {19-01 (19-02)}, + MRNUMBER = {1282290}, +MRREVIEWER = {Dominique Arlettaz}, + DOI = {10.1007/978-1-4612-4314-4}, + URL = {https://doi.org/10.1007/978-1-4612-4314-4}, +} + +# Rosenberg-Schochet +@article {RosenbergSchochet87, + AUTHOR = {Rosenberg, Jonathan and Schochet, Claude}, + TITLE = {The {K}\"{u}nneth theorem and the universal coefficient theorem + for {K}asparov's generalized {$K$}-functor}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {55}, + YEAR = {1987}, + NUMBER = {2}, + PAGES = {431--474}, + ISSN = {0012-7094}, + MRCLASS = {46L80 (19K33 46M20 58G12)}, + MRNUMBER = {894590}, +MRREVIEWER = {Thierry Fack}, + DOI = {10.1215/S0012-7094-87-05524-4}, + URL = {https://doi.org/10.1215/S0012-7094-87-05524-4}, +} + + +### SSSSSSSSSSSSSSSSSSS +# Sakai +@article {Sakai55, + AUTHOR = {Sakai, Sh\^{o}ichir\^{o}}, + TITLE = {On the group isomorphism of unitary groups in {AW}*-algebras}, + JOURNAL = {Tohoku Math. J. (2)}, + FJOURNAL = {The Tohoku Mathematical Journal. Second Series}, + VOLUME = {7}, + YEAR = {1955}, + PAGES = {87--95}, + ISSN = {0040-8735}, + MRCLASS = {46.2X}, + MRNUMBER = {73139}, +MRREVIEWER = {J. Feldman}, + DOI = {10.2748/tmj/1178245106}, + URL = {https://doi.org/10.2748/tmj/1178245106}, +} + +@article {Sakai70, + AUTHOR = {Sakai, Sh\^{o}ichir\^{o}}, + TITLE = {An uncountable number of {II}$_1$ and {II}$_{\infty}$ factors}, + JOURNAL = {J. Functional Analysis}, + VOLUME = {5}, + YEAR = {1970}, + PAGES = {236--246}, + MRCLASS = {46.65}, + MRNUMBER = {0259626}, +MRREVIEWER = {Z. Takeda}, + DOI = {10.1016/0022-1236(70)90028-5}, + URL = {https://doi.org/10.1016/0022-1236(70)90028-5}, +} + +@book{Sakaibook, + title={{C}*-algebras and {W}*-algebras}, + author={Sakai, Sh\^{o}ichir\^{o}}, + year={2012}, + publisher={Springer Science \& Business Media} +} + +# Sarkowicz +@article{Sarkowicz23inclusion, + title={Tensorially Absorbing Inclusions of {C}*-algebras}, + author={Pawel Sarkowicz}, + year={2023}, + eprint={2211.14974}, + archivePrefix={arXiv}, + primaryClass={math.OA}, + journal={arXiv:2211.14974}, + NOTE={preprint} +} + +@article{Sarkowicz23unitary, + title={Unitary groups, {K}-theory and traces}, + author={Pawel Sarkowicz}, + year={2023}, + eprint={2305.15989}, + archivePrefix={arXiv}, + primaryClass={math.OA}, + journal={arXiv:2305.15989}, + NOTE={preprint} +} + +# Sarkowicz-Tikuisis +@article{SarkowiczTikuisis, + title={Polar decomposition in algebraic {K}-theory}, + author={Pawel Sarkowicz and Aaron Tikuisis}, + year={2023}, + eprint={2303.16248}, + archivePrefix={arXiv}, + primaryClass={math.OA}, + journal={arXiv:2303.16248}, + NOTE={preprint} +} + +# Sato +@article {Sato10, + AUTHOR = {Sato, Yasuhiko}, + TITLE = {The {R}ohlin property for automorphisms of the {J}iang-{S}u + algebra}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {259}, + YEAR = {2010}, + NUMBER = {2}, + PAGES = {453--476}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L35 46L40 46L80)}, + MRNUMBER = {2644109}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.1016/j.jfa.2010.04.006}, + URL = {https://doi.org/10.1016/j.jfa.2010.04.006}, +} + +# Sato-White-Winter +@article {SatoWhiteWinter15, + AUTHOR = {Sato, Yasuhiko and White, Stuart and Winter, Wilhelm}, + TITLE = {Nuclear dimension and {$\mathcal{Z}$}-stability}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {202}, + YEAR = {2015}, + NUMBER = {2}, + PAGES = {893--921}, + ISSN = {0020-9910}, + MRCLASS = {46L35 (45L05)}, + MRNUMBER = {3418247}, +MRREVIEWER = {Aaron Tikuisis}, + DOI = {10.1007/s00222-015-0580-1}, + URL = {https://doi.org/10.1007/s00222-015-0580-1}, +} + +# Schafhauser +@article {Schafhauser20, + AUTHOR = {Schafhauser, Christopher}, + TITLE = {Subalgebras of simple {AF}-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {192}, + YEAR = {2020}, + NUMBER = {2}, + PAGES = {309--352}, + ISSN = {0003-486X}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {4151079}, +MRREVIEWER = {Daniele Puglisi}, + DOI = {10.4007/annals.2020.192.2.1}, + URL = {https://doi.org/10.4007/annals.2020.192.2.1}, +} + +# Schemaitat +@article {Schemaitat22, + AUTHOR = {Schemaitat, Andr\'{e}}, + TITLE = {The {J}iang-{S}u algebra is strongly self-absorbing revisited}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {282}, + YEAR = {2022}, + NUMBER = {6}, + PAGES = {Paper No. 109347, 39}, + ISSN = {0022-1236}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {4360358}, +MRREVIEWER = {Prahlad Vaidyanathan}, + DOI = {10.1016/j.jfa.2021.109347}, + URL = {https://doi.org/10.1016/j.jfa.2021.109347}, +} + +# Schochet +@article {SchochetIV, + AUTHOR = {Schochet, Claude}, + TITLE = {Topological methods for {C}*-algebras. {IV}. {M}od {$p$} homology}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {114}, + YEAR = {1984}, + NUMBER = {2}, + PAGES = {447--468}, + ISSN = {0030-8730}, + MRCLASS = {46L80 (19K33 46M20 55N99)}, + MRNUMBER = {757511}, +MRREVIEWER = {Vern Paulsen}, + URL = {http://projecteuclid.org/euclid.pjm/1102708718}, +} + +# Serre +@book {Serre77, + AUTHOR = {Serre, Jean-Pierre}, + TITLE = {Linear representations of finite groups}, + SERIES = {Graduate Texts in Mathematics, Vol. 42}, + NOTE = {Translated from the second French edition by Leonard L. Scott}, + PUBLISHER = {Springer-Verlag, New York-Heidelberg}, + YEAR = {1977}, + PAGES = {x+170}, + ISBN = {0-387-90190-6}, + MRCLASS = {20CXX}, + MRNUMBER = {0450380}, +MRREVIEWER = {W. Feit}, +} + +# Shoda +@article {Shoda37, + AUTHOR = {Shoda, Kenjiro}, + TITLE = {Einige {S}\"{a}tze \"{u}ber {M}atrizen}, + JOURNAL = {Jpn. J. Math.}, + FJOURNAL = {Japanese Journal of Mathematics}, + VOLUME = {13}, + YEAR = {1937}, + NUMBER = {3}, + PAGES = {361--365}, + ISSN = {0075-3432}, + MRCLASS = {15A24}, + MRNUMBER = {3223061}, + DOI = {10.4099/jjm1924.13.0\_361}, + URL = {https://doi.org/10.4099/jjm1924.13.0_361}, +} + +# Silverman +@book{Silverman14, + title={A friendly introduction to number theory}, + author={Silverman, Joseph H.}, + year={2014}, + publisher={Pearson} +} + +# Suziki +@article {Suzuki21, + AUTHOR = {Suzuki, Yuhei}, + TITLE = {Equivariant $\mathcal{O}_2$-absorption theorem for exact groups}, + JOURNAL = {Compos. Math.}, + FJOURNAL = {Compositio Mathematica}, + VOLUME = {157}, + YEAR = {2021}, + NUMBER = {7}, + PAGES = {1492--1506}, + ISSN = {0010-437X,1570-5846}, + MRCLASS = {46L55 (46L05)}, + MRNUMBER = {4275465}, +MRREVIEWER = {Qingzhai\ Fan}, + DOI = {10.1112/s0010437x21007168}, + URL = {https://doi.org/10.1112/s0010437x21007168}, +} + +@article {Szabo18, + AUTHOR = {Szab\'{o}, G\'{a}bor}, + TITLE = {Equivariant {K}irchberg-{P}hillips-type absorption for + amenable group actions}, + JOURNAL = {Comm. Math. Phys.}, + FJOURNAL = {Communications in Mathematical Physics}, + VOLUME = {361}, + YEAR = {2018}, + NUMBER = {3}, + PAGES = {1115--1154}, + ISSN = {0010-3616,1432-0916}, + MRCLASS = {37A55 (46L40)}, + MRNUMBER = {3830263}, +MRREVIEWER = {Francesco\ Fidaleo}, + DOI = {10.1007/s00220-018-3110-3}, + URL = {https://doi.org/10.1007/s00220-018-3110-3}, +} + +# Sutherland-Takesaki +@article {SutherlandTakesaki89, + AUTHOR = {Sutherland, Colin E. and Takesaki, Masamichi}, + TITLE = {Actions of discrete amenable groups on injective factors of + type {${\rm III}_\lambda,\;\lambda\neq 1$}}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {137}, + YEAR = {1989}, + NUMBER = {2}, + PAGES = {405--444}, + ISSN = {0030-8730,1945-5844}, + MRCLASS = {46L40 (22D25 46L10 46L55)}, + MRNUMBER = {990219}, +MRREVIEWER = {Sze-Kai\ Tsui}, + URL = {http://projecteuclid.org/euclid.pjm/1102650391}, +} + +### TTTTTTTTTTTTTTTTTTTT + +# Takoutsing-Robert +@article{TakoutsingRobert23, + title={Distinguishing {C}*-algebras by their unitary groups}, + author={Takoutsing, Lionel Fogang and Robert, Leonel}, + journal={arXiv:2306.15825}, + year={2023} +} + +# Tatsuuma-Shimomura-Hirai +@article {TatsuumaShimomuraHirai98, + AUTHOR = {Tatsuuma, Nobuhiko and Shimomura, Hiroaki and Hirai, Takeshi}, + TITLE = {On group topologies and unitary representations of inductive + limits of topological groups and the case of the group of + diffeomorphisms}, + JOURNAL = {J. Math. Kyoto Univ.}, + FJOURNAL = {Journal of Mathematics of Kyoto University}, + VOLUME = {38}, + YEAR = {1998}, + NUMBER = {3}, + PAGES = {551--578}, + ISSN = {0023-608X}, + MRCLASS = {22A05 (54H11 58D05)}, + MRNUMBER = {1661157}, +MRREVIEWER = {M. Rajagopalan}, + DOI = {10.1215/kjm/1250518067}, + URL = {https://doi.org/10.1215/kjm/1250518067}, +} + +# Thomsen +@article {Thomsen91, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Nonstable {$K$}-theory for operator algebras}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {4}, + YEAR = {1991}, + NUMBER = {3}, + PAGES = {245--267}, + ISSN = {0920-3036}, + MRCLASS = {46L80 (19K33 46L85)}, + MRNUMBER = {1106955}, +MRREVIEWER = {Berndt Brenken}, + DOI = {10.1007/BF00569449}, + URL = {https://doi.org/10.1007/BF00569449}, +} + +@article {Thomsen93, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Finite sums and products of commutators in inductive limit {C}*-algebras}, + JOURNAL = {Ann. Inst. Fourier (Grenoble)}, + FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier}, + VOLUME = {43}, + YEAR = {1993}, + NUMBER = {1}, + PAGES = {225--249}, + ISSN = {0373-0956}, + MRCLASS = {46L05 (46M40)}, + MRNUMBER = {1209702}, +MRREVIEWER = {Guihua Gong}, + URL = {http://www.numdam.org/item?id=AIF_1993__43_1_225_0}, +} + +@article {Thomsen95, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Traces, unitary characters and crossed products by {$\mathbb{Z}$}}, + JOURNAL = {Publ. Res. Inst. Math. Sci.}, + FJOURNAL = {Kyoto University. Research Institute for Mathematical + Sciences. Publications}, + VOLUME = {31}, + YEAR = {1995}, + NUMBER = {6}, + PAGES = {1011--1029}, + ISSN = {0034-5318}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {1382564}, +MRREVIEWER = {Kevin McClanahan}, + DOI = {10.2977/prims/1195163594}, + URL = {https://doi.org/10.2977/prims/1195163594}, +} + + +@article {Thomsen97, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Limits of certain subhomogeneous {C}*-algebras}, + JOURNAL = {M\'{e}m. Soc. Math. Fr. (N.S.)}, + FJOURNAL = {M\'{e}moires de la Soci\'{e}t\'{e} Math\'{e}matique de France. Nouvelle S\'{e}rie}, + NUMBER = {71}, + YEAR = {1997}, + PAGES = {vi+125 pp. (1998)}, + ISSN = {0249-633X}, + MRCLASS = {46L05 (46L80 46M20)}, + MRNUMBER = {1649315}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.24033/msmf.385}, + URL = {https://doi.org/10.24033/msmf.385}, +} + +@misc{Thomsen22, + title={On weights, traces and {K}-theory}, + author={Klaus Thomsen}, + year={2022}, + eprint={2211.03172}, + archivePrefix={arXiv}, + primaryClass={math.OA} +} + +# Tikuisis +@article {Tikuisis12, + AUTHOR = {Tikuisis, Aaron}, + TITLE = {Regularity for stably projectionless, simple {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {263}, + YEAR = {2012}, + NUMBER = {5}, + PAGES = {1382--1407}, + ISSN = {0022-1236}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {2943734}, +MRREVIEWER = {Hiroyuki Osaka}, + DOI = {10.1016/j.jfa.2012.05.020}, + URL = {https://doi.org/10.1016/j.jfa.2012.05.020}, +} + +@article {Tikuisis16, + AUTHOR = {Tikuisis, Aaron}, + TITLE = {{K}-theoretic characterization of {C}*-algebras with approximately inner flip}, + JOURNAL = {Int. Math. Res. Not. IMRN}, + FJOURNAL = {International Mathematics Research Notices. IMRN}, + YEAR = {2016}, + NUMBER = {18}, + PAGES = {5670--5694}, + ISSN = {1073-7928}, + MRCLASS = {46L35 (19K99)}, + MRNUMBER = {3567256}, +MRREVIEWER = {Cristian Ivanescu}, + DOI = {10.1093/imrn/rnv334}, + URL = {https://doi.org/10.1093/imrn/rnv334}, +} + + +# Tikuisis-White-Winter +@article {TikuisisWhiteWinter17, + AUTHOR = {Tikuisis, Aaron and White, Stuart and Winter, Wilhelm}, + TITLE = {Quasidiagonality of nuclear {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {185}, + YEAR = {2017}, + NUMBER = {1}, + PAGES = {229--284}, + ISSN = {0003-486X}, + MRCLASS = {46L05 (47L40)}, + MRNUMBER = {3583354}, +MRREVIEWER = {Dinesh Jayantilal Karia}, + DOI = {10.4007/annals.2017.185.1.4}, + URL = {https://doi.org/10.4007/annals.2017.185.1.4}, +} + +# Toms +@article {Toms05, + AUTHOR = {Toms, Andrew S.}, + TITLE = {On the independence of {$K$}-theory and stable rank for simple + {C}*-algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {578}, + YEAR = {2005}, + PAGES = {185--199}, + ISSN = {0075-4102}, + MRCLASS = {46L80 (19K33 46L05)}, + MRNUMBER = {2113894}, +MRREVIEWER = {Nadia S. Larsen}, + DOI = {10.1515/crll.2005.2005.578.185}, + URL = {https://doi.org/10.1515/crll.2005.2005.578.185}, +} + +@article {Toms08, + AUTHOR = {Toms, Andrew S.}, + TITLE = {On the classification problem for nuclear {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {167}, + YEAR = {2008}, + NUMBER = {3}, + PAGES = {1029--1044}, + ISSN = {0003-486X}, + MRCLASS = {46L35 (19K14 46L80)}, + MRNUMBER = {2415391}, +MRREVIEWER = {Francesc Perera}, + DOI = {10.4007/annals.2008.167.1029}, + URL = {https://doi.org/10.4007/annals.2008.167.1029}, +} + +@article {Toms08b, + AUTHOR = {Toms, Andrew S.}, + TITLE = {An infinite family of non-isomorphic {C}*-algebras with + identical {$K$}-theory}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {360}, + YEAR = {2008}, + NUMBER = {10}, + PAGES = {5343--5354}, + ISSN = {0002-9947}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {2415076}, +MRREVIEWER = {Wilhelm Winter}, + DOI = {10.1090/S0002-9947-08-04583-2}, + URL = {https://doi.org/10.1090/S0002-9947-08-04583-2}, +} + +# Toms-Winter +@article {TomsWinter07, + AUTHOR = {Toms, Andrew S. and Winter, Wilhelm}, + TITLE = {Strongly self-absorbing {C}*-algebras}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {359}, + YEAR = {2007}, + NUMBER = {8}, + PAGES = {3999--4029}, + ISSN = {0002-9947}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {2302521}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.1090/S0002-9947-07-04173-6}, + URL = {https://doi.org/10.1090/S0002-9947-07-04173-6}, +} + +@article {TomsWinter08, + AUTHOR = {Toms, Andrew S. and Winter, Wilhelm}, + TITLE = {{$\mathcal{Z}$}-stable {ASH} algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {60}, + YEAR = {2008}, + NUMBER = {3}, + PAGES = {703--720}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (46L35 46L80)}, + MRNUMBER = {2414961}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.4153/CJM-2008-031-6}, + URL = {https://doi.org/10.4153/CJM-2008-031-6}, +} + +@article {TomsWinter09, + AUTHOR = {Toms, Andrew S. and Winter, Wilhelm}, + TITLE = {The {E}lliott conjecture for {V}illadsen algebras of the first + type}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {256}, + YEAR = {2009}, + NUMBER = {5}, + PAGES = {1311--1340}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {2490221}, +MRREVIEWER = {Xiao Chun Fang}, + DOI = {10.1016/j.jfa.2008.12.015}, + URL = {https://doi.org/10.1016/j.jfa.2008.12.015}, +} + +# Toms-White-Winer +@article {TomsWhiteWinter15, + AUTHOR = {Toms, Andrew S. and White, Stuart and Winter, Wilhelm}, + TITLE = {$\mathcal{Z}$-stability and finite-dimensional tracial + boundaries}, + JOURNAL = {Int. Math. Res. Not. IMRN}, + FJOURNAL = {International Mathematics Research Notices. IMRN}, + YEAR = {2015}, + NUMBER = {10}, + PAGES = {2702--2727}, + ISSN = {1073-7928}, + MRCLASS = {46L35 (46L40)}, + MRNUMBER = {3352253}, +MRREVIEWER = {C. J. K. Batty}, + DOI = {10.1093/imrn/rnu001}, + URL = {https://doi.org/10.1093/imrn/rnu001}, +} + +### UUUUUUUUUUUUUUUU + +### VVVVVVVVVVVVVVVVV +# Viladsen +@article {Villadsen98, + AUTHOR = {Villadsen, Jesper}, + TITLE = {Simple {C}*-algebras with perforation}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {154}, + YEAR = {1998}, + NUMBER = {1}, + PAGES = {110--116}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {1616504}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.1006/jfan.1997.3168}, + URL = {https://doi.org/10.1006/jfan.1997.3168}, +} + +@article {Villadsen99, + AUTHOR = {Villadsen, Jesper}, + TITLE = {On the stable rank of simple {C}*-algebras}, + JOURNAL = {J. Amer. Math. Soc.}, + FJOURNAL = {Journal of the American Mathematical Society}, + VOLUME = {12}, + YEAR = {1999}, + NUMBER = {4}, + PAGES = {1091--1102}, + ISSN = {0894-0347}, + MRCLASS = {46L05 (18B10 19K99 46L80 46M20)}, + MRNUMBER = {1691013}, +MRREVIEWER = {Vicumpriya S. Perera}, + DOI = {10.1090/S0894-0347-99-00314-8}, + URL = {https://doi.org/10.1090/S0894-0347-99-00314-8}, +} + +# Voiculescu +@article{Voiculescu83, + title={Asymptotically commuting finite rank unitary operators without commuting approximants}, + author={Voiculescu, Dan}, + journal={Acta Sci. Math.(Szeged)}, + volume={45}, + number={1-4}, + pages={429--431}, + year={1983} +} +@article {Voiculescu83, + AUTHOR = {Voiculescu, Dan}, + TITLE = {Asymptotically commuting finite rank unitary operators without + commuting approximants}, + JOURNAL = {Acta Sci. Math. (Szeged)}, + FJOURNAL = {Acta Universitatis Szegediensis. Acta Scientiarum + Mathematicarum}, + VOLUME = {45}, + YEAR = {1983}, + NUMBER = {1-4}, + PAGES = {429--431}, + ISSN = {0001-6969}, + MRCLASS = {47B44}, + MRNUMBER = {708811}, +} + +### WWWWWWWWWWWWWWWWWWWWWWWWWWWW +# Wiebel +@book {Kbook, + AUTHOR = {Weibel, Charles A.}, + TITLE = {The {$K$}-book}, + SERIES = {Graduate Studies in Mathematics}, + VOLUME = {145}, + NOTE = {An introduction to algebraic $K$-theory}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {2013}, + PAGES = {xii+618}, + ISBN = {978-0-8218-9132-2}, + MRCLASS = {19-01}, + MRNUMBER = {3076731}, +MRREVIEWER = {L. N. Vaserstein}, + DOI = {10.1090/gsm/145}, + URL = {https://doi.org/10.1090/gsm/145}, +} + +# Willett-Yu +@article{WillettYu21, + title={The {UCT} for {C}*-algebras with finite complexity}, + author={Willett, Rufus and Yu, Guoliang}, + journal={arXiv e-prints}, + pages={arXiv--2104}, + year={2021} +} + +# Winter +@article {Winter11, + AUTHOR = {Winter, Wilhelm}, + TITLE = {Strongly self-absorbing {C}*-algebras are $\mathcal{Z}$-stable}, + JOURNAL = {J. Noncommut. Geom.}, + FJOURNAL = {Journal of Noncommutative Geometry}, + VOLUME = {5}, + YEAR = {2011}, + NUMBER = {2}, + PAGES = {253--264}, + ISSN = {1661-6952}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {2784504}, +MRREVIEWER = {Camillo Trapani}, + DOI = {10.4171/JNCG/74}, + URL = {https://doi.org/10.4171/JNCG/74}, +} + +@article {Winter12, + AUTHOR = {Winter, Wilhelm}, + TITLE = {Nuclear dimension and $\mathcal{Z}$-stability of pure {C}*-algebras}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {187}, + YEAR = {2012}, + NUMBER = {2}, + PAGES = {259--342}, + ISSN = {0020-9910}, + MRCLASS = {46L85 (46L35)}, + MRNUMBER = {2885621}, + DOI = {10.1007/s00222-011-0334-7}, + URL = {https://doi.org/10.1007/s00222-011-0334-7}, +} + +### XXXXXXXXXXXXXXXXXXXXX + +### YYYYYYYYYYYYYYYYYYYYY + +# Yen +@article {Yen56, + AUTHOR = {Yen, Ti}, + TITLE = {Isomorphism of unitary groups in {AW}*-algebras}, + JOURNAL = {Tohoku Math. J. (2)}, + FJOURNAL = {The Tohoku Mathematical Journal. Second Series}, + VOLUME = {8}, + YEAR = {1956}, + PAGES = {275--280}, + ISSN = {0040-8735}, + MRCLASS = {46.2X}, + MRNUMBER = {89376}, +MRREVIEWER = {I. E. Segal}, + DOI = {10.2748/tmj/1178244951}, + URL = {https://doi.org/10.2748/tmj/1178244951}, +} + +### ZZZZZZZZZZZZZZZZZZZZ + + diff --git a/Unitary groups, K-theory and traces/letterfonts.tex b/Unitary groups, K-theory and traces/letterfonts.tex new file mode 100644 index 0000000..87fcf4f --- /dev/null +++ b/Unitary groups, K-theory and traces/letterfonts.tex @@ -0,0 +1,81 @@ +%commands +\newcommand{\ms}{\mathscr} +\newcommand{\mc}{\mathcal} +\newcommand{\mf}{\mathfrak} +\newcommand{\nin}{\not\in} +\newcommand{\bs}{\backslash} +\newcommand{\nsg}{\unlhd} +\newcommand{\ov}{\overline} +%blackboard +\newcommand{\bN}{\mathbb{N}} +\newcommand{\bR}{\mathbb{R}} +\newcommand{\bZ}{\mathbb{Z}} +\newcommand{\bQ}{\mathbb{Q}} +\newcommand{\bF}{\mathbb{F}} +\newcommand{\bK}{\mathbb{K}} +\newcommand{\bG}{\mathbb{G}} +\newcommand{\bE}{\mathbb{E}} +\newcommand{\bC}{\mathbb{C}} +\newcommand{\bV}{\mathbb{V}} +\newcommand{\bA}{\mathbb{A}} +\newcommand{\bP}{\mathbb{P}} +\newcommand{\bD}{\mathbb{D}} +\newcommand{\bT}{\mathbb{T}} +\newcommand{\bM}{\mathbb{M}} +\newcommand{\bB}{\mathbb{B}} +\newcommand{\bL}{\mathbb{L}} +\newcommand{\bX}{\mathbb{X}} +\newcommand{\bY}{\mathbb{Y}} +\newcommand{\bI}{\mathbb{I}} + + +\newcommand{\1}{{\bf 1}} +\newcommand{\0}{{\bf 0}} + +%mathcal +\newcommand{\cM}{\mathcal{M}} +\newcommand{\cN}{\mathcal{N}} +\newcommand{\cC}{\mathcal{C}} +\newcommand{\cB}{\mathcal{B}} +\newcommand{\cS}{\mathcal{S}} +\newcommand{\cI}{\mathcal{I}} +\newcommand{\cO}{\mathcal{O}} +\newcommand{\cP}{\mathcal{P}} +\newcommand{\cA}{\mathcal{A}} +\newcommand{\cL}{\mathcal{L}} +\newcommand{\cF}{\mathcal{F}} +\newcommand{\cH}{\mathcal{H}} +\newcommand{\cK}{\mathcal{K}} +\newcommand{\cD}{\mathcal{D}} +\newcommand{\cE}{\mathcal{E}} +\newcommand{\cT}{\mathcal{T}} +\newcommand{\cZ}{\mathcal{Z}} +\newcommand{\cU}{\mathcal{U}} +\newcommand{\cJ}{\mathcal{J}} +\newcommand{\cG}{\mathcal{G}} +\newcommand{\cR}{\mathcal{R}} +\newcommand{\cQ}{\mathcal{Q}} +\newcommand{\cY}{\mathcal{Y}} +%terns +\newcommand{\bh}{\mathcal{B}(\mathcal{H})} +\newcommand{\bk}{\mathcal{B}(\mathcal{K})} +\newcommand{\kh}{\mathcal{K}(\mathcal{H})} +\newcommand{\fh}{\mathcal{F}(\mathcal{H})} +\newcommand{\sa}{\mathcal{S}(\mathcal{A})} +\newcommand{\bx}{\mathcal{B}(\mathbb{X}} +\newcommand{\by}{\mathcal{C}(\mathbb{Y}} + +%frak +\newcommand{\fX}{\mathfrak{X}} +\newcommand{\fY}{\mathfrak{Y}} +\newcommand{\fS}{\mathfrak{S}} +\newcommand{\fM}{\mathfrak{M}} +\newcommand{\fN}{\mathfrak{N}} +\newcommand{\fF}{\mathfrak{F}} +\newcommand{\fJ}{\mathfrak{J}} +\newcommand{\fT}{\mathfrak{T}} +\newcommand{\fA}{\mathfrak{A}} +\newcommand{\fB}{\mathfrak{B}} +\newcommand{\fP}{\mathfrak{P}} +\newcommand{\fQ}{\mathfrak{Q}} +\newcommand{\fO}{\mathfrak{O}} diff --git a/Unitary groups, K-theory and traces/macros.tex b/Unitary groups, K-theory and traces/macros.tex new file mode 100644 index 0000000..14414c5 --- /dev/null +++ b/Unitary groups, K-theory and traces/macros.tex @@ -0,0 +1,108 @@ +% tensor products +\newcommand{\thk}{\cH\otimes\cK} +\newcommand{\amb}{\bA \otimes_{\max} \bB} +\newcommand{\omax}{\otimes_{\max}} +\newcommand{\omin}{\otimes_{\min}} +\newcommand{\ovn}{\overline{\otimes}} + +% probability +\newcommand{\PS}{(\Omega,\cM,\cP)} +\newcommand{\rvf}{f:\Omega \to \bR} +\newcommand{\linf}{\cL^{\infty -}\PS} +\newcommand{\probc}{\text{Prob}_c(\bR)} +\newcommand{\prob}{\text{Prob}(\bR)} + +% maps +\newcommand{\into}{\hookrightarrow} +\newcommand{\onto}{\twoheadrightarrow} +\newcommand{\act}{\curvearrowright} + +\newcommand{\ee}{\varepsilon} +\newcommand{\sm}{\setminus} +\newcommand{\Om}{\Omega} +\newcommand{\simi}{\sim_{\infty}} + +%limits +\newcommand{\limsupn}{\underset{n}{\text{limsup}}} +\newcommand{\sotc}{\stackrel{\text{SOT}}{\to}} +\newcommand{\wotc}{\stackrel{\text{WOT}}{\to}} +\newcommand{\sotlim}{\text{SOT-}\lim} +\newcommand{\wotlim}{\text{WOT-}\lim} + +% useful for nuclear, lifting maps, etc.... +\newcommand{\tphi}{\tilde{\phi}} +\newcommand{\tpsi}{\tilde{\psi}} +\newcommand{\hphi}{\hat{\phi}} +\newcommand{\hpsi}{\hat{\psi}} +\newcommand{\phil}{\phi_\lambda} +\newcommand{\psil}{\psi_\lambda} +\newcommand{\mkl}{M_{k(\lambda)}} +\newcommand{\mkn}{M_{k(n)}} +\newcommand{\phin}{\phi_n} +\newcommand{\psin}{\psi_n} +\newcommand{\vphi}{\varphi} + +\newcommand{\floor}[1]{\lfloor #1 \rfloor} + +\newcommand{\supp}{\text{supp}} +\newcommand{\wk}{\text{wk}} +\newcommand{\Tr}{\text{Tr}} +\newcommand{\dist}{\text{dist}} +\newcommand{\rank}{\text{rank}} +\newcommand{\sgn}{\text{sgn}} + +\newcommand{\ev}{\text{ev}} +\newcommand{\spr}{\text{spr}} +\newcommand{\tD}{\tilde{\Delta}} +\newcommand{\uv}{\underline} +\newcommand{\tr}{\text{tr}} +\newcommand{\id}{\text{id}} +\newcommand{\ad}{\text{ad}} +\newcommand{\Ad}{\text{Ad}} + +\newcommand*{\tc}[2]{(\ov{#1}^{#2},#2)} + +% Types for von Neumann algebras +\newcommand{\I}{\text{I}} +\newcommand{\II}{\text{II}} +\newcommand{\III}{\text{III}} +\newcommand{\IIIl}{\text{III}_{\lambda}} +% note the last character of this last one is an "ell", its just that my font has capital i and lower-case l being the same character..... + + +%operators +% basic algebra +\DeclareMathOperator{\Ann}{Ann} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\im}{im} + +% K-theory and regularity for C*-algebras +\DeclareMathOperator{\Ell}{Ell} +\DeclareMathOperator{\Aff}{Aff} +\DeclareMathOperator{\KT}{KT} +\DeclareMathOperator{\Cu}{Cu} +\DeclareMathOperator{\ka}{K^{\text{alg}}} +\DeclareMathOperator{\ku}{K^\text{alg,u}} +\DeclareMathOperator{\hka}{\ov{K}_1^{\text{alg}}} +\DeclareMathOperator{\hku}{\ov{K}_1^{\text{alg,u}}} + +\newcommand{\MvN}{\sim_{\text{MvN}}} + + +% exponential rank and length for C*-algebras +\newcommand{\cer}{\text{cer}} +\newcommand{\cel}{\text{cel}} + +% KK-theory and Ext +\DeclareMathOperator{\KK}{KK} +\DeclareMathOperator{\KKn}{KK_{\text{nuc}}} +\DeclareMathOperator{\KKs}{KK_{\text{sep}}} +\DeclareMathOperator{\Ext}{Ext} +\DeclareMathOperator{\Extn}{Ext_{\text{nuc}}} + + +% direct/inverse limits +\newcommand{\dlim}{\underset{\to}{\lim}} +\newcommand{\ilim}{\underset{\leftarrow}{\lim}} diff --git a/Unitary groups, K-theory and traces/preabmle.tex b/Unitary groups, K-theory and traces/preabmle.tex new file mode 100644 index 0000000..ae186b4 --- /dev/null +++ b/Unitary groups, K-theory and traces/preabmle.tex @@ -0,0 +1,53 @@ +% change this depending on what you're doing. +%\documentclass[11pt]{amsart} + +%packages +%\usepackage[margin=1.5in]{geometry} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[pdftex]{graphicx} +\usepackage{tikz-cd} +\usepackage{blindtext} + +\usepackage{hyperref,xcolor} +\usepackage{amsmath,amsfonts,amssymb,amsthm,color,mathtools,enumitem} +\usepackage{epstopdf} +\usepackage{polynom} +\usepackage{babel} +\usepackage{mathrsfs} +\usepackage{chngcntr} +\usepackage{verbatim} + +\hypersetup{ + colorlinks=true, + linkcolor=blue, + citecolor=cyan, + urlcolor=cyan} + + \usepackage{hyphenat} + + %theorems +\newtheorem{theorem}{Theorem}[section] +\newtheorem{defn}[theorem]{Definition} +\newtheorem{prop}[theorem]{Proposition} +\newtheorem{cor}[theorem]{Corollary} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{example}[theorem]{Example} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{question}[theorem]{Question} + +\newtheorem*{resultA}{Theorem A} +\newtheorem*{resultB}{Theorem B} +\newtheorem*{resultC}{Theorem C} +\newtheorem*{resultD}{Theorem D} +\newtheorem*{resultE}{Theorem E} +\newtheorem*{resultF}{Theorem F} + +%\newtheorem{result}{Theorem}[chapter] +\newtheorem{result}{Theorem} +%\renewcommand*{\theresult}{\arabic{chapter}.\Alph{result}} +\renewcommand*{\theresult}{\Alph{result}} +\newtheorem{resultcor}[result]{Corollary} + +\numberwithin{equation}{section} + diff --git a/Unitary groups, K-theory and traces/unitary_group_homs.bbl b/Unitary groups, K-theory and traces/unitary_group_homs.bbl new file mode 100644 index 0000000..e48d6b5 --- /dev/null +++ b/Unitary groups, K-theory and traces/unitary_group_homs.bbl @@ -0,0 +1,202 @@ +\newcommand{\etalchar}[1]{$^{#1}$} +\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} +\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } +% \MRhref is called by the amsart/book/proc definition of \MR. +\providecommand{\MRhref}[2]{% + \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} +} +\providecommand{\href}[2]{#2} +\begin{thebibliography}{ARBG12} + +\bibitem[AM03]{AraMathieubook} +Pere Ara and Martin Mathieu, \emph{Local multipliers of {C}*-algebras}, + Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, + 2003. \MR{1940428} + +\bibitem[ARBG12]{AlBoothGiordano12} +Ahmed Al-Rawashdeh, Andrew Booth, and Thierry Giordano, \emph{Unitary groups as + a complete invariant}, J. Funct. Anal. \textbf{262} (2012), no.~11, + 4711--4730. \MR{2913684} + +\bibitem[Boo98]{Booth98} +Andrew Booth, \emph{The unitary group as a complete invariant for simple unital + {AF} algebras}, Master's thesis, University of Ottawa (Canada), 1998. + +\bibitem[Bre93]{Bresar93} +Matej Bre\v{s}ar, \emph{Commuting traces of biadditive mappings, + commutativity-preserving mappings and {L}ie mappings}, Trans. Amer. Math. + Soc. \textbf{335} (1993), no.~2, 525--546. \MR{1069746} + +\bibitem[CET{\etalchar{+}}21]{CETWW21} +Jorge Castillejos, Samuel Evington, Aaron Tikuisis, Stuart White, and Wilhelm + Winter, \emph{Nuclear dimension of simple {C}*-algebras}, Invent. Math. + \textbf{224} (2021), no.~1, 245--290. \MR{4228503} + +\bibitem[CGS{\etalchar{+}}23]{CGSTW23} +Jos{\'e}~R. Carri{\'o}n, James Gabe, Christopher Schafhauser, Aaron Tikuisis, + and Stuart White, \emph{Classifying *-homomorphisms {I}: Unital simple + nuclear {C}*-algebras}, arXiv:2307.06480 (2023). + +\bibitem[Con85]{Conway19} +John~B. Conway, \emph{A course in functional analysis}, Graduate Texts in + Mathematics, vol.~96, Springer-Verlag, New York, 1985. \MR{768926} + +\bibitem[CP79]{CuntzPedersen79} +Joachim Cuntz and Gert~K. Pedersen, \emph{Equivalence and traces on + {C}*-algebras}, Journal of Functional Analysis \textbf{33} (1979), no.~2, + 135--164. + +\bibitem[CR23]{ChandRobert23} +Abhinav Chand and Leonel Robert, \emph{Simplicity, bounded normal generation, + and automatic continuity of groups of unitaries}, Adv. Math. \textbf{415} + (2023), Paper No. 108894, 52. \MR{4543451} + +\bibitem[Dad95]{Dadarlat95} +Marius Dadarlat, \emph{Reduction to dimension three of local spectra of real + rank zero {C}*-algebras}, J. Reine Angew. Math. \textbf{460} (1995), + 189--212. \MR{1316577} + +\bibitem[dlH13]{dlHarpe13} +Pierre de~la Harpe, \emph{Fuglede--{K}adison determinant: theme and + variations}, Proc. Natl. Acad. Sci. USA \textbf{110} (2013), no.~40, + 15864--15877. \MR{3363445} + +\bibitem[dlHS84a]{dlHS84a} +Pierre de~la Harpe and Georges Skandalis, \emph{D\'{e}terminant associ\'{e} \`a + une trace sur une alg\'{e}bre de {B}anach}, Ann. Inst. Fourier (Grenoble) + \textbf{34} (1984), no.~1, 241--260. \MR{743629} + +\bibitem[dlHS84b]{dlHS84b} +\bysame, \emph{Produits finis de commutateurs dans les {C}*-alg\`ebres}, Ann. + Inst. Fourier (Grenoble) \textbf{34} (1984), no.~4, 169--202. \MR{766279} + +\bibitem[Dye53]{Dye53} +H.~A. Dye, \emph{The unitary structure in finite rings of operators}, Duke + Math. J. \textbf{20} (1953), 55--69. \MR{52695} + +\bibitem[Dye55]{Dye55} +\bysame, \emph{On the geometry of projections in certain operator algebras}, + Ann. of Math. 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Can. \textbf{42} (2020), no.~4, + 451--539. \MR{4215380} + +\bibitem[Gon97]{Gong97} +Guihua Gong, \emph{On inductive limits of matrix algebras over + higher-dimensional spaces. {I}, {II}}, Math. Scand. \textbf{80} (1997), + no.~1, 41--55, 56--100. \MR{1466905} + +\bibitem[Goo86]{Goodearlbook} +Kenneth~R. Goodearl, \emph{Partially ordered abelian groups with + interpolation}, Mathematical Surveys and Monographs, vol.~20, American + Mathematical Society, Providence, RI, 1986. \MR{845783} + +\bibitem[GS16]{GiordanoSierakowski16} +Thierry Giordano and Adam Sierakowski, \emph{The general linear group as a + complete invariant for {C}*-algebras}, J. 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J. + (2) \textbf{8} (1956), 275--280. \MR{89376} + +\end{thebibliography} diff --git a/Unitary groups, K-theory and traces/unitary_group_homs.tex b/Unitary groups, K-theory and traces/unitary_group_homs.tex new file mode 100644 index 0000000..9bca68f --- /dev/null +++ b/Unitary groups, K-theory and traces/unitary_group_homs.tex @@ -0,0 +1,1029 @@ +\documentclass[12pt]{amsart} + +\usepackage[margin=1.5in]{geometry} + +\include{preabmle} +\include{macros} +\include{letterfonts} + +\begin{document} +\title{Unitary groups, $K$-theory and traces} +\author{Pawel Sarkowicz} +%\email{\href{mailto:email}{email}} +\email{\href{mailto:psark007@uottawa.ca}{psark007@uottawa.ca}} +\address{Department of Mathematics and Statistics, University of Ottawa, 75 Laurier Ave. East, Ottawa, ON, K1N 6N5 Canada} + + \begin{abstract} + We show that continuous group homomorphisms between uni\hyp{}tary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain $K$-theoretic regularity conditions, these maps can be seen to commute with the pairing between $K_0$ and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign). + \end{abstract} + + \maketitle + \tableofcontents + + \section{Introduction} + + + %\section{Introduction}\label{section:ugh-introduction} +\renewcommand*{\thetheorem}{\arabic{chapter}.\Alph{theorem}} +Unitary groups of C*-algebras have been long studied, and for many classes of operator algebras they form a complete invariant. In \cite{Dye53}, Dye studied the unitary group isomorphism problem between non-atomic W*-algebras, with the assumption of \emph{weak bicontinuity} of the isomorphism. He later showed that the unitary group, this time as an algebraic object, determined the type of a factor \cite{Dye55} (except for type $\I_{2n}$). He showed that such group isomorphisms were the restrictions of a *-isomorphism or a conjugate linear *-isomorphism multiplied by a possibly discontinuous character (\cite[Appendix A]{Booth98} gives exposition). Sakai generalized Dye's results to show that any uniformly continu\hyp{}ous unitary group isomorphism between AW*-factors comes from a *-isomor\hyp{}phism or conjugate-linear *-isomorphism \cite{Sakai55} (see also \cite{Yen56} for general AW*-algebras which have no component of type $I_n$). + +Dye's method was generalized to large classes of real rank zero C*-algebras by Al-Rawashdeh, Booth and Giordano in \cite{AlBoothGiordano12}, where they applied the method to obtain induced maps between $K$-theory, with a general linear variant being done by Giordano and Sierakowski in \cite{GiordanoSierakowski16}. The stably finite and purely infinite cases were handled separately. The unital, simple AH-algebras of slow dimension growth and of real rank zero were classified by the topological group isomorphism class of their unitary groups (or general linear groups), and the unital, simple, purely infinite UCT algebras were classified via the algebraic isomorphism classes of their unitary groups (or general linear groups). These results made use of the abundance of projections in real rank zero C*-algebras (at least to show there were isomorphic $K_0$-groups), and made use of the Dadarlat-Elliott-Gong \cite{Dadarlat95,Gong97} and Kirchberg-Phillips \cite{Phillips00} classification theorems respectively (see Theorems 3.3.1 and 8.4.1 of \cite{RordamBook} for each respective case). + +In \cite{Paterson83}, it was proven by Paterson that two unital C*-algebras are iso\hyp{}morphic if and only there is an isometric isomorphism of the unitary groups which acts as the identity on the circle. In a similar vein, the metric structure of the unitary group has also played a role in determining the Jordan *-algebra structure on C*-algebras. In \cite{HatoriMolnar14}, Hatori and Moln{\'a}r showed that two unital C*-algebras are Jordan *-isomorphic if and only if their unitary groups are isometric as metric spaces, not taking into account any algebraic structure. + +Chand and Robert have shown in \cite{ChandRobert23} that if $A$ and $B$ are prime traceless C*-algebras with full square zero elements such that $U^0(A)$, the subgroup of unitaries which are path connected to the identity, is algebraically isomorphic to $U^0(B)$, then $A$ is either isomorphic or anti-isomorphic to $B$. In fact, the group isomorphism is the restriction of a *-isomorphism or anti-*-isomorphism which follows from the fact that unitary groups associated to these C*-algebras have certain automatic continuity properties that allow one to use character\hyp{}izations of \emph{commutativity preserving maps} \cite{Bresar93} (see \cite{AraMathieubook}). Chand and Robert also show that if $A$ is a unital separable C*-algebra with at least one tracial state, then $U^0(A)$ admits discontinuous automorphisms. Thus the existence of traces is an obstruction to classification via algebraic structure on the unitary groups -- at least an obstruction to unitary group homomorphisms being the restrictions of *-homomorphisms or anti-*-homomorphisms. + +In this paper, we show that uniformly continuous unitary group homo\hyp{}morphisms yield maps between traces which have several desirable $K$-theoretic properties -- especially under stricter continuity assumptions. Namely, that the homomorphism sends the circle to the circle and is contractive, which would be automatic if it had a lift to a *-homomorphism or conjugate-linear *-homomorphism. + +We state our main results. Recall that if $A$ is a unital C*-algebra, $T(A)$ denotes the simplex of tracial states and $\Aff T(A)$ is the real Banach space of continuous affine functions $T(A) \to \bR$. + +\begin{result}[Corollary \ref{cor:pairing}] +Let $A,B$ be unital C*-algebras. If $\theta: U^0(A) \to U^0(B)$ is a uniformly continuous group homomorphism, then there exists a bounded $\bR$-linear map $\Lambda_\theta: \Aff T(A) \to \Aff T(B)$ such that +\begin{equation} +\begin{tikzcd} +\pi_1(U^0(A)) \arrow[r, "\tD^1_A"] \arrow[d, "\pi_1(\theta)"'] & \Aff T(A) \arrow[d, "\Lambda_\theta"] \\ +\pi_1(U^0(B)) \arrow[r, "\tD^1_B"'] & \Aff T(B) +\end{tikzcd} +\end{equation} +commutes. +\end{result} + +Here $\pi_1(\theta)$ is the map between fundamental groups induced by $\theta$, and, for a C*-algebra $A$, $\tD^1_A$ is the \emph{pre-determinant} (used in the definition of the de la Harpe-Skandalis determinant associated to the universal trace) that takes a piece-wise smooth path in $U^0(A)$ +%\footnote{As every continuous path of unitaries (resp. invertibles) is homotopic to a piece-wise smooth path of unitaries (resp. invertibles) -- see Proposition \ref{rem:cts-to-piece-wise-smooth}(1) -- and $\tD^1_A$ is homotopy-invariant, one can apply $\tD^1_A$ to any path of unitaries (resp. invertibles).} +beginning at the unit to an element of $\Aff T(A)$. See Section \ref{section:dlHS-det} for details. + +Recall that the $K_0$-group of a unital C*-algebra can be identified with the fundamental group $\pi_1(U_{\infty}^0(A))$. Restricting to C*-algebras with sufficient $K_0$-regularity -- by this we mean C*-algebras whose $K_0$-group can be realized as loops in the connected component of its unitary group -- we get a map between $K_0$-groups and a map between spaces of continuous real-valued affine functions on the trace simplex which commute with the pairing. + +\begin{resultcor}[Corollary \ref{cor:pairing}] +Let $A,B$ be unital C*-algebras such that the canonical maps +\begin{equation}\label{eq:cor3-hyp-iso} +\pi_1(U^0(A)) \to K_0(A) \text{ and }\pi_1(U^0(B)) \to K_0(B) +\end{equation} +are isomorphisms. If $\theta: U^0(A) \to U^0(B)$ is a continuous group homomorphism then there exists a bounded linear map $\Lambda_\theta: \Aff T(A) \to \Aff T(B)$ such that +\begin{equation} +\begin{tikzcd} +K_0(A) \arrow[r, "\rho_A"] \arrow[d, "K_0(\theta)"'] & \Aff T(A) \arrow[d, "\Lambda_\theta"] \\ +K_0(B) \arrow[r, "\rho_B"'] & \Aff T(B) +\end{tikzcd} +\end{equation} +commutes, where $K_0(\theta)$ is the map induced by $\pi_1(\theta)$ together with the isomorphisms of (\ref{eq:cor3-hyp-iso}). +\end{resultcor} + +C*-algebras satisfying the above hypothesis are quite common -- for example C*-algebras having stable rank one \cite{Rieffel87} or that are $\cZ$-stable \cite{Jiang97} have this property. Viewing $\Aff T(A)$ and $\Aff T(B)$ as partially ordered real Banach spaces (under the uniform norm) with order units, it is not however true that $\Lambda_\theta$ is unital or positive (see Example \ref{example:nonpositive-example}). This is remedied by adding stricter continuity assumptions on the homomorphism $\theta$ (and possibly by replacing $\Lambda_\theta$ with $-\Lambda_\theta$). + +When $\theta: U(A) \to U(B)$ is contractive, injective and sends the circle to the circle, then we show (Lemma \ref{lem:tildeS-isometric}) that either $\Lambda_\theta$ or $-\Lambda_\theta$ is unital and positive, and therefore $\theta$ induces a map between $K$-theory and traces in such a manner that respects the pairing (which in turn gives a map between Elliott invariants for certain simple C*-algebras). As a consequence, we can identify certain unitary subgroups with C*-subalgebras by using $K$-theoretic classification of embeddings \cite{CGSTW23}. + +\begin{result}[Corollary \ref{cor:ugh-subgroup-subalgebra}] +Let $A$ be a unital, separable, simple, nuclear C*-algebra satisfying the UCT which is either $\cZ$-stable or has stable rank one, and $B$ be a unital, separable, simple, nuclear $\cZ$-stable C*-algebra. If there is a contractive injective group homomorphism $U(A) \to U(B)$ which maps the circle to the circle, then there is a unital embedding $A \into B$. +\end{result} + +This paper is structured as follows. In Section \ref{section:ugh-ctugh-and-traces} we use a continuous unitary group homomorphism to construct a map between spaces of continuous affine functions on the trace simplices, and use the de la Harpe-Skandalis determinant to show that this map has several desirable properties with respect to the map induced on the fundamental groups of the unitary groups. In Section \ref{section:ugh-order-on-aff-classification} we discuss how our map between spaces of affine functions respects or flips the order under certain continuity assumptions on the unitary group homomorphism. In Section \ref{section:ugh-general-linear-variants} we discuss a slight general linear variant. We finish in Section \ref{section:ugh-final-remarks} with some open questions. + +%\addtocontents{toc}{\protect\setcounter{tocdepth}{0}} +%\section*{Acknowledgements} +%Many thanks to my supervisors Thierry Giordano and Aaron Tikuisis for many helpful discussions. + %Acknowledgements section + \addtocontents{toc}{\protect\setcounter{tocdepth}{0}} + \section*{Acknowledgements} + Many thanks to my PhD supervisors Thierry Giordano and Aaron Tikuisis for many helpful discussions. Thanks to the authors of \cite{CGSTW23} for sharing a draft of their paper. Finally, thanks to the referee for asking for clarification on the general linear variant, which led me to consider a counter-example to $\bC$-linearity. + + %go back to normal TOC ordering.. + \addtocontents{toc}{\protect\setcounter{tocdepth}{1}} + +%\addtocontents{toc}{\protect\setcounter{tocdepth}{2}} +\renewcommand*{\thetheorem}{\arabic{section}.\arabic{theorem}} + + +\section{Preliminaries and notation} + +\subsection{Notation} +%We will use capital letters $A,B,C,D,\dots$ to denote C*-algebras. Generally small letters $a,b,c,d,\dots,x,y,z$ will denote operators in C*-algebras. +For a group $G$, we denote by $DG$ the derived subgroup of $G$, i.e., +\begin{equation} DG := \langle ghg^{-1}h^{-1} \mid g,h \in G \rangle. \end{equation} +If $G$ has an underlying topology, we denote by $CG$ the closure of $DG$ and $G^0$ the connected component of the identity. + +For a unital C*-algebra $A$, $U(A)$ denotes the unitary group of $A$, while $U^0(A)$ denotes the connected component of the identity in $U(A)$. For $n \in \bN$, we write $U_n(A) := U(M_n(A))$, $U_n^0(A) := U^0(M_n(A))$, and we set +\begin{equation} U_{\infty}(A) := \dlim\, U_n(A), \end{equation} +to be the inductive limit with connecting maps $U_n(A) \ni u \mapsto u \oplus 1 \in U_{n+1}(A)$. +This makes $U_\infty(A)$ both a topological space (with the inductive limit topology) and a group.\footnote{As pointed out in footnote 58 of \cite{CGSTW23}, $U_{\infty}(A)$ is not in general a topological group since multiplication is not in general jointly continuous in this topology.} +We have general linear analogues by replacing $U$ with $GL$, where $GL(A)$ denotes the group of invertible elements of $A$. +Similarly, we define $M_{\infty}(A) = \dlim \, M_n(A)$ (as an algebraic direct limit) with connecting maps $x \mapsto x \oplus 0$. If $E$ is real Banach space and $\tau: A_{sa} \to E$ is a linear map that is tracial (i.e., $\tau(a^*a) = \tau(aa^*)$ for all $a \in A$), we extend this canonically to $(M_{\infty}(A))_{sa}$ by setting $\tau((a_{ij})) := \sum_i \tau(a_{ii})$ for $(a_{ij})\in (M_n(A)_{sa}$. + +We write $\pi_0(X)$ for the space of connected components of a topological space $X$, and $\pi_1(X)$ for the fundamental group of $X$ with distinguished base point. In our case, we will usually have $X = U_n(A)$ or $X = U_n^0(A)$, for $n \in \bN \cup \{\infty\}$, with the base point being the unit. + +For a unital C*-algebra $A$, we let $K_0(A),K_1(A)$ be the topological $K$-groups of $A$. +We will often use the identification of $K_0(A)$ with the fundamental group $\pi_1(U_{\infty}^0(A))$ (see for example \cite[Chapter 11.4]{RordamKBook}). +%If $A$ is unital, $K_0(A)$ is the Grothendieck group of the Murray-von Neumann semigroup of projections in $M_{\infty}(A)$ and +%\begin{equation} K_1(A) = U_{\infty}/U_{\infty}^0(A) \simeq GL_{\infty}/GL_{\infty}^0(A). \end{equation} +The set of tracial states on $A$ will be denoted $T(A)$, which is a Choquet simplex (\cite[Theorem 3.1.18]{Sakaibook}), and we denote by $\Aff T(A)$ the set of continuous affine functions $T(A) \to \bR$, which is an interpolation group with order unit (see \cite[Chapter 11]{Goodearlbook}). +%The space $\Aff T(A)$ can also be viewed as a topological group under addition, equipped with the uniform norm topology. +For unital $A$, the pairing map $\rho_A: K_0(A) \to \Aff T(A)$ is defined as follows: if $x \in K_0(A)$, we can write $x = [p] - [q]$ where $p,q \in M_n(A)$ are projections, and then +\begin{equation} +\rho_A(x)(\tau) := \tau (p - q), \ \ \ \tau \in T(A). +\end{equation} + +\subsection{The de la Harpe--Skandalis determinant and Thomsen's variant}\label{section:dlHS-det} + +We recall the definition of the unitary variant of the de la Harpe--Skandalis determinant \cite{dlHS84a} (see \cite{dlHarpe13} for a more in-depth exposition). By a bounded trace we mean a bounded linear map $\tau: A_{sa} \to E$, where $E$ is a real Banach space, such that $\tau(a^*a) = \tau(aa^*)$ for all $a \in A$. For $n \in \bN \cup \{\infty\}$, a bounded trace $\tau: A_{sa} \to E$, and a piece-wise smooth path $\xi: [0,1] \to U_n(A)$, set +\begin{equation} +\label{eq:predetDefn} +\tD_\tau^n(\xi) := \int_0^1 \tau\left(\frac{1}{2\pi i}\xi'(t)\xi(t)^{-1}\right)dt, +\end{equation} +where this integral is just the Riemann integral in $E$.\footnote{Note that $\xi'(t)\xi(t)$ is skew-adjoint by \cite[Proposition 1.4]{Phillips92} so that $\frac{1}{2\pi i}\xi'(t)\xi(t)^{-1}$ is self-adjoint.} + We state the unitary variant of \cite[Lemme 1]{dlHS84a}. + +\begin{prop}\label{prop:detFacts} +Let $\tau: A_{sa} \to E$ be a bounded trace and $n \in \bN \cup \{\infty\}$. The map $\tD_\tau^n$, which takes a piece-wise smooth path in $U_n^0(A)$ to an element in $E$, has the following four properties: + \begin{enumerate} + \item it takes pointwise products to sums: if $\xi_1,\xi_2$ are two piece-wise smooth paths, then + \begin{equation} \tD_\tau^n(\xi_1\xi_2) = \tD_\tau^n(\xi_1) + \tD_\tau^n(\xi_2), \end{equation} + where $\xi_1\xi_2$ is the piece-wise smooth path $t \mapsto \xi_1(t)\xi_2(t)$ from $\xi_1(0)\xi_2(0)$ to $\xi_1(1)\xi_2(1)$; + \item if $\|\xi(t) - 1\| < 1$ for all $t \in [0,1]$, then \begin{equation} 2\pi i \tD_\tau^n(\xi) = \tau\big(\log\xi(1) - \log\xi(0)\big); \end{equation} + \item $\tD_\tau^n(\xi)$ depends only on the continuous homotopy class of $\xi$; + \item if $p \in M_n(A)$ is a projection, then the path $\xi_p: [0,1] \to U_n^0(A)$ given by $\xi_p(t) := pe^{2\pi i t} + (1-p)$ satisfies $\tD_\tau^n(p) = \tau(p)$. \label{predetprojection} + \end{enumerate} +\end{prop} + + The de la Harpe--Skandalis determinant associated to $\tau$ (at the $n^{\text{th}}$ level) is then the map + \begin{equation} \Delta_\tau^n: U_{\infty}^0(A) \to E/\tD_\tau^n(\pi_1(U_n^0(A))) \end{equation} + given by $\Delta_\tau^n(x) := [\tD_\tau^n(\xi_x)]$ where $\xi_x$ is any piece-wise smooth path $\xi_x: [0,1] \to U_n^0(A)$ from 1 to $x$. This is a well-defined group homomorphism (using Proposition \ref{prop:detFacts}) to an abelian group and therefore factors through the derived group, i.e., $DU_n^0(A) \subseteq \ker \Delta_\tau^n$. For the $n = \infty$ case, we just write $\tD_\tau$ and $\Delta_\tau$ for $\tD_\tau^{\infty}$ and $\Delta_\tau^{\infty}$ respectively. + + We will often be interested in the \emph{universal trace} $\Tr_A: A_{sa} \to \Aff T(A)$, which is given by $\Tr_A(a) := \hat{a}$, where $\hat{a} \in \Aff T(A)$ is the function given by $\hat{a}(\tau) := \tau(a)$ for $\tau \in T(A)$. We note that in this case, for $[x] \in K_0(A)$, we have that $\Tr(x) = \rho_A([x])$. When considering the universal trace, we will write $\Delta^n$ and $\Delta$ for $\Delta_{\Tr}^n$ and $\Delta_{\Tr}^{\infty}$ respectively. If the C*-algebra needs to be specified, we write $\Delta_A^n$ or $\Delta_A$. + + \begin{prop}\label{rem:cts-to-piece-wise-smooth}\mbox{} +Let $n \in \bN \cup \{\infty\}$. Every continuous path $\xi: [0,1] \to U_n(A)$ is homotopic to a piece-wise smooth path {\normalfont (}even a piece-wise smooth exponential path if we are in $U_n^0(A)${\normalfont )}. Moreover, there exists $a \in A_{sa}$ such that $\tD_\omega^n(\xi) = \omega(a)$ whenever $\omega: A_{sa} \to E$ is a bounded trace. + +In particular, as $\tD^n$ is homotopy-invariant, it makes sense to apply $\tD^n$ to any continuous path. +\begin{proof} +This is essentially \cite[Lemme 3]{dlHS84a}. Take a continuous path $\xi: [0,1] \to U_n(A)$ and choose $k$ such that + \begin{equation} + \left\|\xi(\frac{j-1}{k})^{-1}\xi(\frac{j}{k}) - 1\right\| < 1 \text{ for all } j=1,\dots,k. + \end{equation} + Then taking + \begin{equation} + a_j := \frac{1}{2\pi i}\log \left(\xi(\frac{j-1}{k})^{-1}\xi(\frac{j}{k})\right), j=1,\dots,k, + \end{equation} + $\xi$ will be homotopic to the path + \begin{equation} + \eta(t) = \xi\left(\frac{j-1}{k}\right)e^{2\pi i(kt - j+1)a_j}, t \in \left[\frac{j-1}{k},\frac{j}{k}\right], j=1,\dots,k. + \end{equation} + We note that $\tD_\tau^n(\eta) = \sum_{j=1}^k \tau(a_j)$. Indeed, for simplicity denote by + \begin{equation} + X_j := \xi\left(\frac{j-1}{k}\right)\text{ and }Y_j := e^{2\pi i(kt - j + 1)a_j}. + \end{equation} + Then +\begin{equation}\label{eq:xi-to-eta-ajs} +\begin{split} + \tD_\omega^n(\xi) &= \tD_\omega^n(\eta) \\ + &=\sum_{j=1}^k \int_{\frac{j-1}{k}}^{\frac{j}{k}}\tau\left(\frac{1}{2\pi i}X_j 2\pi i k a_jY_jY_j^*X_j^*\right) dt \\ + &= \sum_{j=1}^k \int_{\frac{j-1}{k}}^{\frac{j}{k}}k\tau(a_j) dt \\ + &= \sum_{j=1}^k \tau(a_j). \\ + \end{split} +\end{equation} +If we take $a := \tr(a')$, where $a' = \sum_{j=1}^k a_j$ (here $\tr: M_{\infty}(A) \to A$ is the unnormalized trace), we see that $\tD_\omega^n(\xi) = \omega(a)$. + \end{proof} + \end{prop} + + Let $A_0$ consist of elements $a \in A_{sa}$ satisfying $\tau(a) = 0$ for all $\tau \in T(A)$. + This is a norm-closed real subspace of $A_{sa}$ such that $A_0 \subseteq \ov{[A,A]}$, and there is an isometric identification $A_{sa}/A_0 \simeq \Aff T(A)$ sending an element $[a]$ to $\widehat{a}$. + Indeed, it is not hard to see that the map $A_{sa}/A_0 \to \Aff T(A)$ given by $[a] \mapsto \hat{a}$ is a well-defined $\bR$-linear map. + Moreover \cite[Theorem 2.9]{CuntzPedersen79}, together with a convexity argument, gives that this is isometric identification. + To see that we have all the real-valued affine functions, we note that the image of this map contains constant functions and separates points, so \cite[Corollary 7.4]{Goodearlbook} gives that the image is dense and therefore all of $\Aff T(A)$ (since this is an isometry). + We freely identify $A_{sa}/A_0$ with $\Aff T(A)$. + + \subsection{Thomsen's variant} Thomsen's variant of the de la Harpe--Skandalis determinant is the Hausdorffized version, taking into account the closure of the image of the homotopy groups. For a bounded trace $\tau: A_{sa} \to E$, we consider the map + \begin{equation} \bar{\Delta}_\tau^n: U_n^0(A) \to E/\ov{\tD_\tau^n(\pi_1(U_n^0(A)))}, \end{equation} + given by $\bar{\Delta}_\tau^n(x) := [\tD_\tau^n(\xi_x)]$ where $\xi_x: [0,1] \to U_n^0(A)$ is any piece-wise smooth path from 1 to $x \in U_n^0(A)$. +This is similar to the map $\Delta_\tau^n$, except the codomain is now the quotient by the closure of the image of the fundamental group under the pre-determinant (i.e., the Hausdorffization of the codomain). +When considering the universal trace, we just write $\ov\Delta^n$ for $\ov\Delta_{\Tr}^n$ and $\ov\Delta$ for $\ov\Delta_{\Tr}^{\infty}$. If the C*-algebra needs to be specified, we write $\ov\Delta_A^n$ or $\ov\Delta_A$. + +If one considers the universal trace, the kernel of $\ov\Delta^n$ can be identified with $CU_n^0(A)$ (where the closure is taken with respect to the inductive limit topology in the $n = \infty$ case). + +\begin{lemma}[Lemma 3.1, \cite{Thomsen95}] +Let $A$ be a unital C*-algebra. Then +\begin{equation} +\ker \ov\Delta^n = CU_n^0(A). +\end{equation} +\end{lemma} + +It is not in general true that the kernel of $\Delta^n$ can be identified with the derived group $DU_n^0(A)$, although there are several positive results \cite{dlHS84b,Thomsen93,Ng14,NgRobert17,NgRobert15}. + +It immediately follows that the quotient of $U_n^0(A)$ by the closure of the commutator subgroup (under the inductive limit topology in the $n = \infty$ case) can be identified with a quotient of $\Aff T(A)$. + +\begin{theorem}[Theorem 3.2, \cite{Thomsen95}]\label{theorem:thomsen-iso} +$\ov\Delta^n$ gives a homeomorphic group iso\hyp{}morphism +\begin{equation} +U_n^0(A)/CU_n^0(A) \simeq \Aff T(A)/\ov{\tD^n(\pi_1(U_n^0(A)))} +\end{equation} +for every $n \in \bN \cup \{\infty\}$. In particular, +\begin{equation} +U_{\infty}^0(A)/CU_{\infty}^0(A) \simeq \Aff T(A)/\ov{\rho_A(K_0(A))}. +\end{equation} +\end{theorem} + + +\subsection{The $KT_u$-invariant} + +Following \cite{CGSTW23}, for a unital C*-algebra, we let +\begin{equation} +KT_u(A) := (K_0(A),[1_A]_0,K_1(A),\rho_A,\Aff T(A)) +\end{equation} +be the invariant consisting of the $K_0$-group, the position of the unit in $K_0$, the $K_1$ group, the pairing between $K_0$ and traces, and the continuous real-valued affine functions on the trace simplex (viewed as a partially ordered Banach space with order unit). +For two unital C*-algebras $A,B$, a $KT_u$-morphism +\begin{equation} +(\alpha_0,\alpha_1,\gamma): KT_u(A) \to KT_u(B), +\end{equation} +will be a triple $(\alpha_0,\alpha_1,\gamma)$ consisting of $\alpha_0: K_0(A) \to K_0(B)$ a group homomor\hyp{}phism such that $\alpha_0([1_A]_0) = [1_B]_0$, $\alpha_1: K_1(A) \to K_1(B)$ a group homomorphism, and $\gamma: \Aff T(A) \to \Aff T(B)$ is a unital positive map such that +\begin{equation} +\begin{tikzcd} +K_0(A) \arrow[r, "\rho_A"] \arrow[d, "\alpha_0"'] & \Aff T(A) \arrow[d, "\gamma"] \\ +K_0(B) \arrow[r, "\rho_B"'] & \Aff T(B) +\end{tikzcd} +\end{equation} +commutes. + +We note that for large classes of unital simple C*-algebras -- for example the class of unital, simple, separable, nuclear $\cZ$-stable C*-algebras satisfying the UCT -- $KT_u(\cdot)$ recovers the Elliott invariant. + +\section{Continuous unitary group homomorphisms and traces}\label{section:ugh-ctugh-and-traces} + +\hspace{\parindent}Throughout, $A$ and $B$ will be unital C*-algebras with non-empty trace simplices, and $\theta: U^0(A) \to U^0(B)$ will denote a uniformly continuous group homomorphism between the connected components of the identities in the respective unitary groups. We will specify any additional assumptions as we go along. As $\theta$ is a continuous group homomorphism, it send commutators to commutators and limits of commutators to limits of commutators. Thus there are induced group homomorphisms +\begin{equation} +\begin{split} +&U^0(A)/CU^0(A) \to U^0(B)/CU^0(B) \text{ and }\\ +&U^0(A)/DU^0(A) \to U^0(B)/DU^0(B). +\end{split} +\end{equation} +Thomsen's isomorphism \cite[Theorem 3]{Thomsen95} then brings about maps between quotients of $\Aff T(A)$ and $\Aff T(B)$: +\begin{equation}\label{eq:ch-commute} +\begin{tikzcd} +U^0(A)/CU^0(A) \arrow[r, "\simeq"] \arrow[d] & \Aff T(A)/\ov{\tD_A^1(\pi_1(U^0(A)))} \arrow[d] \\ +U^0(B)/CU^0(B) \arrow[r, "\simeq"'] & \Aff T(B)/\ov{\tD_B^1(\pi_1(U^0(B)))} . +\end{tikzcd} +\end{equation} +In a similar vein, when $DU^0(A) = \ker\Delta_A^1$ and $DU^0(B) = \ker\Delta_B^1$, there is a purely algebraic variant of the above diagram: +\begin{equation}\label{eq:h-commute} +\begin{tikzcd} +U^0(A)/DU^0(A) \arrow[r, "\simeq"] \arrow[d] & \Aff T(A)/\tD_A^1(\pi_1(U^0(A))) \arrow[d] \\ +U^0(B)/DU^0(B) \arrow[r, "\simeq"'] & \Aff T(B)/\tD_B^1(\pi_1(U^0(B))). +\end{tikzcd} +\end{equation} +In fact, there is always a diagram as above with $DU^0(A)$ and $DU^0(B)$ replaced with $\ker\Delta_A^1$ and $\ker\Delta_B^1$, respectively, by Proposition \ref{prop:pre-det-commuting-diagrams}(3). That is, whether or not the kernel of the determinant agrees with the commutator subgroup of the connected component of the identity, we have: +\begin{equation}\label{eq:h-commute-pre-det} +\begin{tikzcd} +U^0(A)/\ker\Delta_A^1 \arrow[r, "\simeq"] \arrow[d] & \Aff T(A)/\tD_A^1(\pi_1(U^0(A))) \arrow[d] \\ +U^0(B)/\ker\Delta_B^1 \arrow[r, "\simeq"'] & \Aff T(B)/\tD_B^1(\pi_1(U^0(B))). +\end{tikzcd} +\end{equation} +The diagram in (\ref{eq:h-commute}) is just a special case of (\ref{eq:h-commute-pre-det}). + + In the setting where $\pi_1(U^0(A)) \to K_0(A)$ and $\pi_1(U^0(B)) \to K_0(B)$ are surjections, we have induced maps between quotients +\begin{equation} +\begin{split} +&\Aff T(A)/\ov{\rho_A(K_0(A))} \to \Aff T(B)/\ov{\rho_B(K_0(B))}, \\ +&\Aff T(A)/\rho_A(K_0(A)) \to \Aff T(B)/\rho_B(K_0(B)) +\end{split} +\end{equation} +in the respective Hausdorffized and non-Hausdorffized settings. + +One question to be answered is whether or not we can lift the maps on the right of (\ref{eq:ch-commute}) and (\ref{eq:h-commute-pre-det}) to maps $\Aff T(A) \to \Aff T(B)$. These spaces have further structure as dimension groups with order units \cite[Chapter 7]{Goodearlbook}, so we would like to be able to alter the lift to get a map which is unital and positive. We show that we can always lift this map, and altering it to be unital and positive is possible under a certain continuity assumption on $\theta$. + +If we further assume that $K_0(A) \simeq \pi_1(U^0(A))$ and $K_0(B) \simeq \pi_1(U^0(B))$ in the canonical way (which is true in the presence of $\cZ$-stability by \cite{Jiang97} or stable rank one \cite{Rieffel87}), we would like this map to be compatible with the group homomorphism +\begin{equation} +K_0(\theta): K_0(A) \to K_0(B) +\end{equation} +arising from the diagram +\begin{equation} +\begin{tikzcd} +K_0(A) \arrow[r, "\simeq"] \arrow[d, "K_0(\theta)"'] & \pi_1(U^0(A)) \arrow[d, "\pi_1(\theta)"] \\ +K_0(B) \arrow[r, "\simeq"'] & \pi_1(U^0(B)). +\end{tikzcd} +\end{equation} +By compatible, we mean that +\begin{equation} +\begin{tikzcd} +K_0(A) \arrow[d, "\pi_1(\theta)"'] \arrow[r, "\rho_A"] & \Aff T(A) \arrow[d] \\ +K_0(B) \arrow[r, "\rho_B"'] & \Aff T(B) +\end{tikzcd} +\end{equation} +commutes, where the map on the right is the lift coming from maps induced by (\ref{eq:ch-commute}) and (\ref{eq:h-commute-pre-det}). If our map between spaces of affine continuous functions is not unital and positive, but we can alter it accordingly, we must do the same to our map between $K_0$. We would still have a commuting diagram as above, but it would give that maps induced on $K_0(\cdot)$ and $\Aff T(\cdot)$ respect the pairing. + +Stone's theorem \cite[Section X.5]{Conway19} allows one to recover from a strongly continuous one parameter family $U(t)$ of unitaries a (possibly unbounded) self-adjoint operator $X$ such that $U(t) = e^{itX}$ for all $t \in \bR$. If it is a norm-continuous one parameter family of unitaries, one can recover a bounded self-adjoint operator $X$, and $X$ will lie in the C*-algebra generated by the unitaries. +The use of Stone's theorem to deduce that continuous group homomorphisms between unitary groups send exponentials to exponentials is not new. Sakai used it in the 1950's in order to show that a norm-continuous group isomorphism between unitary groups of AW*-algebras are induced by a *-isomorphism or conjugate-linear *-isomorphism between the algebras themselves \cite{Sakai55}. More recently, this sort of idea has been used to understand how the metric structure of the unitary groups can be related to the Jordan *-algebra structure of the algebras \cite{HatoriMolnar14}. + +\begin{lemma}\label{lem:stone-theorem-cons} +Let $A,B$ be unital C*-algebras and $\theta: U^0(A) \to U^0(B)$ be a continuous group homomorphism. Suppose that $a \in A_{sa}$ and represent $B \subseteq \bh$ faithfully. Then $(\theta(e^{2\pi i t a}))_{t \in \bR}$ is a one-parameter norm-continuous family of unitaries, and consequently is of the form $(e^{2\pi i t b})_{t \in \bR}$ for a unique $b \in B_{sa}$. +\begin{proof} +Using the fact that $\theta$ is a norm-continuous homomorphism, $t \mapsto \theta(e^{2\pi i t a})$ is a norm-continuous one-parameter family of unitaries. Stone's theorem gives that there is a unique self-adjoint $b \in \bh$ such that $\theta(e^{2\pi i ta}) = e^{2\pi i tb}$ for all $t \in \bR$. The boundedness of $b$ follows from norm-continuity, and the uniqueness follows from the fact that $b = \frac{1}{2\pi i t_0} \log \theta(e^{2\pi i a t_0})$ for sufficiently small $t_0 > 0$. This also gives that $b \in B$ by functional calculus. +\end{proof} +\end{lemma} + +Let $S_\theta: A_{sa} \to B_{sa}$ be defined via the correspondence given above: +\begin{equation} +\theta(e^{2\pi i t a}) = e^{2\pi i t S_\theta(a)} \text{ for all } t \in \bR. +\end{equation} +Then $S_\theta$ is a bounded $\bR$-linear map (see \cite{Sakai55,HatoriMolnar14}, or note that it is easy to see that its kernel is closed). It is also easily checked to respect commutation, and that its canonical extension to a map from $A$ to $B$ actually sends commutators to commutators, although we will not explicitly use this. Recall that for a C*-algebra $A$, $A_0$ denotes the set of self-adjoint elements that vanish on every tracial state. + +\begin{lemma}\label{lem:sa-bijective} +If $\theta:U^0(A) \to U^0(B)$ is a continuous group homomorphism, then $S_\theta$ is a bounded linear map and the following hold. +\begin{enumerate} +\item If $\theta$ is injective, then $S_\theta$ is injective. +\item If $\theta$ is a homeomorphism, then $S_\theta$ is bijective. +\end{enumerate} +\begin{proof} +As already remarked, $S_\theta$ is bounded. Assuming that $\theta$ is injective, suppose that $S_\theta(a) = S_\theta(b)$. Then +\begin{equation} +\theta(e^{2\pi i t a}) = \theta(e^{2\pi i t b}) +\end{equation} +for all $t \in \bR$. Injectivity of $\theta$ gives that that $e^{2\pi i t a} = e^{2\pi i t b}$ for all $t \in \bR$. But this implies that $a = b$ since we can take $t$ appropriately close to 0 and take logarithms. + +Now if we further assume that $\theta$ is a homeomorphism, then $(\theta^{-1}(e^{2\pi i t b}))_{t \in \bR} \subseteq U^0(A)$ is a norm-continuous one-parameter family of unitaries which we can write as $(e^{2\pi i t a})_{t \in \bR}$ for some $a \in A_{sa}$. But then +\begin{equation} +\theta(e^{2\pi i t a}) = \theta \circ \theta^{-1}(e^{\pi i t b}) = e^{2\pi i t b}. +\end{equation} +Thus $S_\theta(a) = b$ by the uniqueness in Lemma \ref{lem:stone-theorem-cons}. +\end{proof} +\end{lemma} + +Recall that a linear map $\tau: A_{sa} \to E$, where $E$ is a real Banach space, is a bounded trace is if it is a bounded $\bR$-linear map such that $\tau(a^*a) = \tau(aa^*)$ for all $a \in A$. Note that this is equivalent to $\tau \circ \Ad_u = \tau$ for all $u \in U(A)$ -- to see this, follow the steps in \cite[Exercise 3.6]{RordamKBook} together with a complexification argument. This is further equivalent to $\tau \circ \Ad_u = \tau$ for all $u \in U^0(A)$ (as one can write every element in a C*-algebra as the linear combination of 4 unitaries, each of which can be made to not have full spectrum). + +\begin{prop}\label{prop:induced-aff-map} +Let $\theta: U^0(A) \to U^0(B)$ be a continuous group homo\hyp{}morphism, $E$ be a real Banach space and $\tau: B_{sa} \to E$ a bounded trace. Then $\tau \circ S_\theta: A_{sa} \to E$ is a bounded trace. In particular, $S_\theta(A_0) \subseteq B_0$ and $S_\theta$ induces a bounded $\bR$-linear map +\begin{equation} +\Lambda_\theta: \Aff T(A) \to \Aff T(B). +\end{equation} +\begin{proof} +Observe that, for $a \in A_{sa}$ and $u \in U^0(A)$, we have +\begin{equation} +\begin{split} +e^{2\pi i t S_\theta(uau^*)} &= \theta(e^{2\pi i t uau^*}) \\ +&= \theta(ue^{2\pi i t a}u^*) \\\ +&= \theta(u)e^{2\pi i t S_\theta(a)}\theta(u)^* \\ +&= e^{2\pi i t \theta(u)S_\theta(a)\theta(u)^*} +\end{split} +\end{equation} +for all $t \in \bR$. Therefore $S_\theta(uau^*) = \theta(u)S_\theta(a)\theta(u)^*$, and applying $\tau$ yields +\begin{equation} +\begin{split} +\tau \circ S_\theta(uau^*) &= \tau\left(\theta(u)S_\theta(a)\theta(u)^*\right) \\ +&= \tau ( S_\theta(a)), +\end{split} +\end{equation} +i.e., $\tau \circ S_\theta$ is tracial. + +Thus if $a \in A_0$, it vanishes on every tracial state (hence on every tracial functional), and so $\tau \circ S_\theta(a) = 0$ for all $\tau \in T(B)$. Therefore $S_\theta(A_0) \subseteq B_0$ and so $S_\theta$ factors through a map +\begin{equation} +\Lambda_\theta: \Aff T(A) \simeq A_{sa}/A_0 \to B_{sa}/B_0 \simeq \Aff T(B), +\end{equation} +where we identify $A_{sa}/A_0 \simeq \Aff T(A)$ and $B_{sa}/B_0 \simeq \Aff T(B)$. +\end{proof} +\end{prop} + +One cannot expect $\Lambda_\theta$ (or $S_\theta$) to be unital or positive, as the following examples show. + +\begin{example}\label{example:nonpositive-example}\mbox{} +\begin{enumerate} +\item Consider a continuous homomorphism $\theta: \bT \to \bT = U^0(\bC) = U(\bC)$. By Pontryagin duality, $\theta(z) = z^n$ for some $n \in \bZ$. We then have that $S_\theta: \bR \to \bR$ is given by $S_\theta(x) = nx$. If $n \neq 1$, clearly $S_\theta$ is not unital. If $n < 0$, then $S_\theta$ is not positive since it sends $1$ to $n < 0$. An important observation, however, is that if $n < 0$, $-S_\theta: \bR \to \bR$ is positive, and $-\frac{1}{n}S$ is unital and positive. +\item Consider $\theta: \bT^3 \to \bT$ given by $\theta(z,w,v) = \ov{z}wv$. The corresponding map $S_\theta: \bR^3 \to \bR$ is given by +\begin{equation} +S_\theta(a,b,c) = -a + b + c. +\end{equation} +Clearly $(1,0,0) \in \bC^3$ is a positive element, however $S_\theta(1,0,0) = -1 < 0$. This map is however unital. +\item Let $\theta: U_2 \to \bT$ be defined by $\theta(u) = \det(u)$. Then $S_\theta: (M_2)_{sa} \to \bR$ is defined by $S_\theta(A) = \tr A$, where $\tr$ is the unnormalized trace. Clearly this map is not unital, but it is positive. +\item Let $\theta: U_2 \to U_3$ be defined by $\theta(u) = u \oplus 1$. Then $S_\theta: (M_2)_{sa} \to (M_3)_{sa}$ is given by $S_\theta(A) = A \oplus 0$, which is again not unital, but is positive. The induced map $\Lambda_\theta: \bR \to \bR$ is given by $\Lambda_\theta(x) = \frac{2}{3}x$ for $x \in \bR$. +\item Let $\theta: \bT \into U_2$ be defined by +\begin{equation} +\theta(\lambda) = \begin{pmatrix} +\lambda \\ & \lambda +\end{pmatrix}. +\end{equation} +Then $S_\theta$ is a unital, positive isometry and $\Lambda_\theta$ gives rise the identity map +\begin{equation} +\bR = \Aff T(\bC) \to \Aff T(M_2) = \bR. +\end{equation} +\item Let $\theta: \bT \into U_2$ be defined by +\begin{equation} +\theta(\lambda) = \begin{pmatrix} +\lambda & \\ & \ov{\lambda} +\end{pmatrix}. +\end{equation} +Then $S_\theta: \bR \to (M_2)_{sa}$ is defined by +\begin{equation} +S_\theta(x) = \begin{pmatrix} +x & \\ & -x +\end{pmatrix} +\end{equation} +and $\Lambda_\theta$ is identically zero. +\end{enumerate} +\end{example} + +The above examples are important. If $\theta(\bT) \subseteq \bT$, which is a moderate assumption (e.g., if $\theta$ was the restriction of a unital *-homomorphism or an anti-*-homomorphism), we can restrict the homomorphism to the circle to get a continuous group homomorphism $\bT \to \bT$. We understand such homomorphisms by Pontryagin duality \cite[Chapter 4]{Follandbook16}. + +We now use (pre-)determinant techniques in order to show desirable relation\hyp{}ships between our maps. + +\begin{prop}\label{prop:pre-det-commuting-diagrams} +Let $A,B$ be unital C*-algebras, $\theta: U^0(A) \to U^0(B)$ be a uniformly continuous homomorphism, $E$ a real Banach space and $\tau: B_{sa} \to E$ a bounded trace. +\begin{enumerate} +\item Let $\xi: [0,1] \to U^0(A)$ be a piece-wise smooth path with $\xi(0) = 1$. Then +\begin{equation} +\tD_{\tau \circ S_\theta}^1(\xi) = \tD_\tau^1(\theta \circ \xi). +\end{equation} +In particular, $\tD_{\tau\circ S_\theta}^1(\pi_1(U^0(A))) \subseteq \tD_\tau^1(\pi_1(U^0(B)))$. +\item The following diagram commutes: +\begin{equation} +\begin{tikzcd} +\pi_1(U^0(A)) \arrow[d, "\pi_1(\theta)"'] \arrow[r, "\tD_{\tau \circ S_\theta}^1"] & E \arrow[d, "\id"] \\ +\pi_1(U^0(B)) \arrow[r, "\tD_\tau^1"'] & E . +\end{tikzcd} +\end{equation} +\item The following diagram commutes: +\begin{equation}\label{eq:det-commutation} +\begin{tikzcd} +U^0(A) \arrow[r, "\Delta_{\tau \circ S_\theta}^1"] \arrow[d, "\theta"'] & E/\tD_{\tau\circ S_\theta}^1(\pi_1(U^0(A))) \arrow[d] \\ +U^0(B) \arrow[r, "\Delta_\tau^1"'] & E/\tD_\tau^1(\pi_1(U^0(B))) , +\end{tikzcd} +\end{equation} +where the map on the right is the canonical map induced from the inclusion $\tD_{\tau \circ S_\theta}^1(\pi_1(U^0(A))) \subseteq \tD_\tau^1(\pi_1(U^0(B)))$ coming from (1). +In particular, $\theta(\ker \Delta_{\tau\circ S_\theta}^1) \subseteq \ker \Delta_\tau^1$. + +The analogous diagram commutes if we consider Thomsen's variant of the de la Harpe-Skandalis determinant associated to $\tau$ and $\tau \circ S_\theta$ in (\ref{eq:det-commutation}). +\end{enumerate} +\begin{proof} +By the proof of Proposition \ref{rem:cts-to-piece-wise-smooth}, we can find $k \in \bN$ and $a_1,\dots,a_k \in (M_n(A))_{sa}$ such that $\xi$ is homotopic to the path $\eta: [0,1] \to U(A)$ given by +\begin{equation}\label{eq:exponential-prod} +\eta(t) = \xi\left(\frac{j-1}{k}\right)e^{2\pi i(kt - j + 1)a_j}, t \in \left[\frac{j-1}{k},\frac{j}{k}\right] +\end{equation} +and we have that +\begin{equation}\label{eq:pre-det-for-all-omega} +\tD_\omega^n(\xi) = \tD_\omega^n(\eta) = \sum_{j=1}^k \omega(a_j) +\end{equation} +whenever $\omega: A_{sa} \to F$ is a bounded trace to a real Banach space $F$. + +Now $\theta \circ \xi$ is homotopic to $\theta \circ \eta$, which has the following form, for $j=1,\dots,k$ and $t \in [\frac{j-1}{k},\frac{j}{k}]$: +\begin{equation} +\begin{split} +\theta \circ \eta(t) &= \theta \circ \xi\left(\frac{j-1}{k}\right)\theta\left(e^{2\pi i (kt - j + 1)a_j}\right) \\ +&= \theta \circ \xi\left(\frac{j-1}{k}\right)e^{2\pi i (kt - j + 1)S_\theta(a_j)}. +\end{split} +\end{equation} +By taking $X_j := \theta \circ \xi\left(\frac{j-1}{k}\right)$ and $Y_j := e^{2\pi i (kt - j + 1)S_\theta(a_j)}$ in (\ref{eq:xi-to-eta-ajs}), we see that +\begin{equation} +\tD_\tau^1(\theta \circ \xi) = \tD_\tau^1(\theta \circ \eta) = \sum_{j=1}^k \tau(S_\theta(a_j)), +\end{equation} +and this last quantity is just $\tD_{\tau \circ S_\theta}^1(\xi)$ by (\ref{eq:pre-det-for-all-omega}). + +As for parts (2) and (3), they follow from (1). Indeed, (2) is obvious and to see (3) we let +\begin{equation} +q: E/\tD_{\tau \circ S_\theta}^1(\pi(U^0(A))) \to E/\tD_\tau^1(\pi(U^0(B))) +\end{equation} +be the canonical surjection. If $\xi_u: [0,1] \to U^0(A)$ is a path from 1 to $u$, then we have that +\begin{equation} +\begin{split} +q(\Delta_{\tau \circ S_\theta}^1(u)) &= q\left(\tD_{\tau \circ S_\theta}^1(\xi_u) + \tD_{\tau \circ S_\theta}^1(\pi_1(U^0(A)))\right) \\ +&= \tD_{\tau \circ S_\theta}^1(\xi_u) + \tD_\tau^1(\pi_1(U^0(B))) \\ +&\overset{(1)}{=} \tD_\tau^1(\theta \circ \xi_u) + \tD_\tau^1(\pi_1(U^0(B))) \\ +&= \Delta_\tau^1(\theta(u)). +\end{split} +\end{equation} + +The remark about Thomsen's variant is obvious. +\end{proof} +\end{prop} + +\begin{cor}\label{cor:pairing} +The following diagram commutes: +\begin{equation}\label{eq:cor:pairing-diagram1} +\begin{tikzcd} +\pi_1(U^0(A)) \arrow[d, "\pi_1(\theta)"'] \arrow[r, "\tD_A^1"] & \Aff T(A) \arrow[d, "\Lambda_\theta"] \\ +\pi_1(U^0(B)) \arrow[r, "\tD_B^1"'] & \Aff T(B). +\end{tikzcd} +\end{equation} +In particular, when the canonical maps +\begin{equation} +\pi_1(U^0(A)) \to K_0(A) \text{ and }\pi_1(U^0(B)) \to K_0(B) +\end{equation} + are isomorphisms, we have that +\begin{equation} +\begin{tikzcd} +K_0(A) \arrow[d, "K_0(\theta)"'] \arrow[r, "\rho_A"] & \Aff T(A) \arrow[d, "\Lambda_\theta"] \\ +K_0(B) \arrow[r, "\rho_B"'] & \Aff T(B) +\end{tikzcd} +\end{equation} +commutes, where $K_0(\theta): K_0(A) \to K_0(B)$ is the map induced from the diagram +\begin{equation} +\begin{tikzcd} +K_0(A) \arrow[r, "\simeq"] \arrow[d, "K_0(\theta)"'] & \pi_1(U^0(A)) \arrow[d, "\pi_1(\theta)"] \\ +K_0(B) \arrow[r, "\simeq"'] & \pi_1(U^0(B)). +\end{tikzcd} +\end{equation} +\begin{proof} +By definition, we have that $\Lambda_\theta \circ \Tr_A = S_\theta \circ \Tr_B$ and so if $\xi: [0,1] \to U^0(A)$ is a (piece-wise smooth) path, we have +\footnotetext[3]{If $\fX,\fY$ are Banach spaces over the same base field, $f: [0,1] \to \fX$ is a Riemann integrable function and $T: \fX \to \fY$ is a bounded linear map, then $T \circ f: [0,1] \to \fY$ is Riemann integrable with $T\left(\int_0^1 f(t)dt\right) = \int_0^1 T \circ f(t) dt$.} +\begin{equation}\label{eq:lambda-S-commutes-with-Tr} +\begin{split} +\Lambda_\theta\left(\tD_{\Tr_A}^1(\xi)\right) &= \Lambda_\theta\left(\int_0^1 \Tr_A\left(\xi'(t)\xi(t)^{-1}\right)dt\right) \\ +&= \int_0^1 \Lambda_\theta \circ \Tr_A\left(\xi'(t)\xi(t)^{-1}\right)dt\ \footnotemark \\ +&= \int_0^1 \Tr_B \circ S_\theta\left(\xi'(t)\xi(t)^{-1}\right)dt \\ +&= \tD_{\Tr_B \circ S_\theta}(\xi). +\end{split} +\end{equation} +Now we note that the diagram +\begin{equation} +\begin{tikzcd} + & \Aff T(A) \arrow[rd, "\Lambda_\theta"] & \\ +\pi_1(U^0(A)) \arrow[ru, "\tD_A^1"] \arrow[d, "\pi_1(\theta)"'] \arrow[rr, "\tD_{\Tr_B \circ S_\theta}^1"] & & \Aff T(B) \arrow[d, "\id"] \\ +\pi_1(U^0(B)) \arrow[rr, "\tD_B^1"'] & & \Aff T(B) +\end{tikzcd} +\end{equation} +commutes, where the top triangle commutes by the (\ref{eq:lambda-S-commutes-with-Tr}) and the bottom square commutes by Proposition \ref{prop:pre-det-commuting-diagrams}(2). This gives that (\ref{eq:cor:pairing-diagram1}) commutes. + +The second part follows since if $\xi$ is a (piece-wise smooth) path in $U_n(A)$ and $m > n$, then +\begin{equation} +\tD_\tau^m(\xi \oplus 1_{m-n}) = \tD_\tau^n(\xi). +\end{equation} +\end{proof} +\end{cor} + + +\begin{prop}\label{prop:S-theta-is-a-lift} +Let $A,B$ be unital C*-algebras. Then the map $\Lambda_\theta$ is a lift of the map +\begin{equation} +\Aff T(A)/\tD_A^1(\pi_1(U^0(A))) \to \Aff T(B)/\tD_B^1(\pi_1(U^0(B))) +\end{equation} +as described in (\ref{eq:h-commute-pre-det}). +\begin{proof} +Let us label the maps in the diagram (\ref{eq:h-commute-pre-det}): +\begin{equation} +\begin{tikzcd} +U^0(A)/\ker \Delta^1_A \arrow[d, "\tilde\theta"'] \arrow[r, "\delta_A^1"] & \Aff T(A)/\tD_A^1(\pi_1(U^0(A))) \arrow[d, "P"] \\ +U^0(B)/\ker \Delta^1_B \arrow[r, "\delta_B^1"'] & \Aff T(B)/\tD_B^1(\pi_1(U^0(B))) +\end{tikzcd} +\end{equation} +where $\tilde\theta([u]) := [\theta(u)]$, $\delta_A^1([e^{2\pi i a}]) := [\hat{a}]$, $\delta_B^1$ is defined similarly, and +\begin{equation} +P = \delta_B^1 \circ \tilde\theta \circ (\delta_A^1)^{-1}. +\end{equation} +But then we have that +\begin{equation} +\begin{split} +P([a]) &= \delta_B^1 \circ \tilde\theta \circ (\delta_A^1)^{-1}([\hat{a}]) \\ +&= \delta_B^1 \circ \tilde\theta([e^{2\pi i a}]) \\ +&= \delta_B^1([\theta(e^{2\pi i a})]) \\ +&= \delta_B^1([e^{2\pi i S_\theta(a)}]) \\ +&= [\widehat{S_\theta(a)}] \\ +&= [\Lambda_\theta(\widehat{a})] +\end{split} +\end{equation} +In particular, the diagram +\begin{equation} +\begin{tikzcd} +\Aff T(A) \arrow[d, "\Lambda_\theta"'] \arrow[r, "q_A^1"] & \Aff T(A)/\tD_A^1(\pi_1(U^0(A))) \arrow[d, "P"] \\ +\Aff T(B) \arrow[r, "q_B^1"'] & \Aff T(B)/\tD_B^1(\pi_1(U^0(B))) +\end{tikzcd} +\end{equation} +commutes, where $q_A^1$ and $q_B^1$ are the respective quotient maps. +\end{proof} +\end{prop} + +\begin{prop}\label{prop:S-theta-is-a-lift-H} +Let $A,B$ be unital C*-algebras and $\theta: U^0(A) \to U^0(B)$ be a continuous group homomorphism. Then $\Lambda_\theta: \Aff T(A) \to \Aff T(B)$ is a lift of the map +\begin{equation} +\Aff T(A)/\ov{\tD_A^1(\pi_1(U^0(A)))} \to \Aff T(B)/\ov{\tD_B^1(\pi_1(U^0(B)))} +\end{equation} +as described in (\ref{eq:ch-commute}). +\begin{proof} +One can mimic the proof above or apply the above result and appeal to the commuting diagram +\begin{equation} +\begin{tikzcd} + \Aff T(A)/\tD_A^1(\pi_1(U^0(A))) \arrow[d] \arrow[r] & \Aff T(A)/\ov{\tD_A^1(\pi_1(U^0(A)))} \arrow[d] \\ +\Aff T(B)/\tD_B^1(\pi_1(U^0(B))) \arrow[r] & \Aff T(B)/\ov{\tD_B^1(\pi_1(U^0(B)))}, +\end{tikzcd} +\end{equation} +where the vertical maps are defined via the diagrams from (\ref{eq:h-commute-pre-det}) and (\ref{eq:ch-commute}) respectively, and the horizontal maps are the canonical surjections. +\end{proof} +\end{prop} + + +In particular, assuming some $K_0$-regularity gives that $\Lambda_\theta$ is a lift of a map between quotients of spaces of continuous real-valued affine functions by images of $K_0$. + +\begin{cor} +Let $A,B$ be unital C*-algebras such that the canonical maps +\begin{equation} +\pi_1(U^0(A)) \to K_0(A) \text{ and }\pi_1(U^0(B)) \to K_0(B) +\end{equation} +are surjections. If $\theta: U^0(A) \to U^0(B)$ is a continuous homomorphism, then $\Lambda_\theta$ is a lift of the maps on the right of the following two commutative diagrams: +\begin{equation} +\begin{tikzcd} +U^0(A)/CU^0(A) \arrow[r, "\simeq"] \arrow[d] & \Aff T(A)/\ov{\rho_A(K_0(A))} \arrow[d] \\ +U^0(B)/CU^0(B) \arrow[r, "\simeq"'] & \Aff T(B)/\ov{\rho_B(K_0(B))} +\end{tikzcd} +\end{equation} +and +\begin{equation} +\begin{tikzcd} +U^0(A)/\ker \Delta_A^1 \arrow[r, "\simeq"] \arrow[d] & \Aff T(A)/\rho_A(K_0(A)) \arrow[d] \\ +U^0(B)/\ker \Delta_B^1\arrow[r, "\simeq"'] & \Aff T(B)/\rho_B(K_0(B)) . +\end{tikzcd} +\end{equation} +Further, if $\ker \Delta_A^1 = DU^0(A)$ and $\ker \Delta_B^1 = DU^0(B)$, then $\Lambda_\theta$ is a lift of the map induced by the diagram +\begin{equation} +\begin{tikzcd} +U^0(A)/DU^0(A) \arrow[r, "\simeq"] \arrow[d] & \Aff T(A)/\rho_A(K_0(A)) \arrow[d] \\ +U^0(B)/DU^0(B)\arrow[r, "\simeq"'] & \Aff T(B)/\rho_B(K_0(B)) . +\end{tikzcd} +\end{equation} +\end{cor} + +C*-algebras satisfying the last condition arise naturally -- for example unital, separable, simple, pure C*-algebras of stable rank 1 such that every 2-quasi\hyp{}tracial state on $A$ is a trace have this property \cite{NgRobert17}. + + +\section{The order on $\Aff T(\cdot)$}\label{section:ugh-order-on-aff-classification} + +\hspace{\parindent}We now examine when the map induced on $\Aff T(\cdot)$ is positive in order to compare $K$-theory, traces, and the pairing. As we saw in Example \ref{example:nonpositive-example}, the map we get between spaces of affine functions on the trace simplices need not be positive nor unital in general. In this section, we will be able to use the map $\Lambda_\theta$ to construct a unital positive map, under some extra assumptions on $\theta$. + +We record the following results as they give us necessary and sufficient conditions for the $\Lambda_\theta$ to be positive. We use the C*-algebra-valued analogue of the fact that any unital, contractive linear functional is positive, along with the fact that completely positive maps are (completely) bounded with the norm determined by the image of the unit. Recall that an operator system is a self-adjoint unital subspace of a C*-algebra. The following is a combination of Proposition 2.11, Theorem 3.9 and Proposition 3.6 in \cite{Paulsenbook}. + +\begin{prop}\label{prop:abelian-operator-systems} +Let $\cS$ be an operator system and $B$ a unital C*-algebra. +\begin{enumerate} +\item If $\phi: \cS \to B$ is a unital contraction, then $\phi$ is positive. \label{unitalcontraction} +\item If $B = C(X)$ and $\phi: \cS \to B$ is positive, then it is bounded with $\|\phi\| = \|\phi(1)\|$. \label{abeliantarget} +\end{enumerate} +\end{prop} + + + +\begin{lemma}\label{lem:inj-st-pm1} +Suppose $A,B$ are unital C*-algebras and $\theta: U^0(A) \to U^0(B)$ be a continuous group homomorphism such that $\theta(\bT) = \bT$. If $\theta|_\bT$ is injective, then $S_\theta(1) \in \{1,-1\}$. +\begin{proof} +The restriction $\theta|_{\bT}: \bT \to \bT$ is a continuous group homomorphism, hence by Pontryagin duality is of the form $\theta(z) = z^n$ for some $n$. Injectivity implies that $n \in \{1,-1\}$. We then have that +\begin{equation} +e^{2\pi i S_\theta(1)t} = \theta(e^{2\pi i t}) = e^{2\pi i nt} +\end{equation} +for all $t \in \bR$. This implies that $S_\theta(1) = n\cdot 1 \in \{1,-1\}$. +\end{proof} +\end{lemma} + +%In the sequel, we will consider $\{0\}$ as a partial ordered Banach space with order unit (the unit being 0, the positive cone being $\{0\}$), and we set $\Aff \emptyset = C(\emptyset) = \{0\}$. With this, assuming that $T(A) \neq \emptyset$ whenever $T(B) \neq \emptyset$, we can speak of the map $\Lambda_\theta: \Aff T(A) \to \Aff T(B)$ be unital and positive, regardless of there being traces. + +In the sequel, we will be interested in the case that $T(A) \neq \emptyset$. The map $S_\theta$ will still descend to a map $\Lambda_\theta: A_{sa}/A_0 \to B_{sa}/B_0$ regardless of their being tracial states on either C*-algebra, but these quotients are zero if there are no traces. If we take the zero Banach space to be partially ordered with order unit trivially, then one can speak of unital and positive maps $\Aff T(A) \to \Aff T(B)$, regardless of whether $T(B)$ is empty or not (any map from a partially ordered Banach space with order unit to the zero Banach space will be unital and positive). However, if $T(A) = \emptyset$ and $T(B) \neq \emptyset$, then our map $\Lambda_\theta$ would have no chance of being unital. We identify the zero Banach space with $\Aff (\emptyset) = C_\bR(\emptyset)$ and with the complex analogues as well. However, the following lemma is true regardless of whether $T(A)$ is empty or not. + +\begin{lemma}\label{lem:tildeS-isometric} +Suppose $A,B$ are unital C*-algebras and $\theta: U^0(A) \to U^0(B)$ is a continuous group homomorphism such that $\theta(\bT) = \bT$. If $\theta$ is injective, the following are equivalent: +\begin{enumerate} +\item one of $\Lambda_\theta$ or $-\Lambda_\theta$ is positive; +\item $\Lambda_\theta$ is contractive. +\end{enumerate} +\begin{proof} +First suppose that $T(A),T(B) \neq \emptyset$. By Lemma \ref{lem:inj-st-pm1}, we know that $S_\theta(1) \in \{1,-1\}$ and consequently $\Lambda_\theta(\hat 1) \in \{\hat{1},\widehat{-1}\}$ (where we recall that, for $a \in A_{sa}$, $\hat{a} \in \Aff T(A)$ is the affine function $\hat{a}(\tau) = \tau(a)$). By replacing $\Lambda_\theta$ with $-\Lambda_\theta$, we can without loss of generality assume that $\Lambda_\theta$ is unital. Using the fact that $\Aff T(A) + i\Aff T(A) \subseteq C(T(A))$ is an operator system and the canonical extension +\begin{equation} +\Lambda_\theta^\bC: \Aff T(A) + i\Aff T(A) \to \Aff T(B) + i\Aff T(B) \subseteq C(T(B)) +\end{equation} +is a unital linear map with abelian target algebra, this is an easy consequence of the two parts of Proposition \ref{prop:abelian-operator-systems}. + +Finally, if $T(A) = \emptyset$, then any map from $\Aff T(A)$ is both contractive and positive trivially. The same is true of any map with codomain $\Aff T(B)$ if $T(B) = \emptyset$. +\end{proof} +\end{lemma} + +\begin{lemma}\label{lem:lipschitzSbounded} +Suppose $A,B$ are unital C*-algebras and $\theta: U^0(A) \to U^0(B)$ is a continuous group homomorphism. Suppose that $T(A) \neq \emptyset$. +\begin{enumerate} +\item $\|\Lambda_\theta\| \leq \|S_\theta\|$. +\item If $K > 0$ is such that $\|\theta(u) - \theta(v)\| \leq K\|u - v\|$ for all $u,v \in U^0(A)$, then $\|S_\theta\| \leq K$ and $\|\Lambda_\theta\| \leq K$. +\item If $\theta$ is a homeomorphism, then $\Lambda_\theta$ is bijective with bounded inverse being $\Lambda_{\theta^{-1}}$. +\item If $\theta$ is isometric, then so is $S_\theta$. +If $\theta$ is a surjective isometry, then $\Lambda_\theta$ is a surjective isometry. +\end{enumerate} + +\begin{proof} +As $S_\theta(A_0) \subseteq B_0$, we have +\begin{equation} +\begin{split} +\|\Lambda_\theta(\hat{a})\|_{\Aff T(B)} &= \|\widehat{S_\theta(a)}\|_{\Aff T(B)} \\ +&= \inf_{b \in B_0} \|S_\theta(a) + b\| \\ +&\leq \inf_{b \in \theta(A_0)} \|S_\theta(a) + b\| \\ +&= \inf_{a' \in A_0} \|S_\theta(a) + S_\theta(a')\| \\ +&\leq \|S_\theta\|\inf_{a' \in A_0}\|a + a'\| \\ +&= \|S_\theta\|\|\hat{a}\|_{\Aff T(A)} +\end{split} +\end{equation} +whenever $a \in A_{sa}$. This gives that $\|\Lambda_\theta\| \leq \|S_\theta\|$. + +The fact that $S_\theta$ is an isometry whenever $\theta$ is an isometry follows from an argument in \cite{HatoriMolnar14}; we exemplify said argument to show the bound condition. We use the observation that +\begin{equation} +\frac{e^{2\pi i t a} - 1}{t} \to 2\pi i a +\end{equation} +as $t \to 0$. Since +\begin{equation} +\|e^{2\pi i t S_\theta(a)} - 1\| \leq K\|e^{2\pi i t a} - 1\| +\end{equation} +for all $t \in \bR$, we can divide both sides by $\frac{1}{2\pi}|t|$ and take $t \to 0$ to get that +\begin{equation} +\|S_\theta(a)\| \leq K\|a\|. +\end{equation} +Thus $\|S_\theta\| \leq K$. It then follows from (1) that $\|\Lambda_\theta\| \leq K$ as well. + +If $\theta$ is a homeomorphism, $S_\theta$ and $S_{\theta^{-1}}$ are both defined and its clear that $S_\theta^{-1} = S_{\theta^{-1}}$. + +The surjectivity of $\theta$ implies the surjectivity of $S_\theta$ and thus if $b \in B_{sa}$, we can find $a \in A_{sa}$ with $S_\theta(a) = b$. But then $\Lambda_\theta(\hat{a}) = \widehat{S_\theta(a)} = \hat{b}$. In particular, $\Lambda_\theta$ is surjective. Now if $\theta$ is a surjective isometry, we identify $\Aff T(A) \simeq A_{sa}/A_0$ and $\Aff T(B) \simeq B_{sa}/B_0$, noting that $S_\theta(A_0) = B_0$, and that $\Lambda_\theta$ will preserve the quotient norms. +\end{proof} +\end{lemma} + +\begin{cor} +Suppose $A,B$ are unital C*-algebras and $\theta: U^0(A) \to U^0(B)$ is a continuous group homomorphism. Suppose that $T(A) \neq \emptyset$. If $S_\theta(1) = n$ and $\|S_\theta\| = |n|$, then $\frac{1}{n}S_\theta$ is a unital contraction, hence positive. In particular, if $\theta(\bT) = \bT$ and $\theta|_{\bT}$ is an injection, then either $\Lambda_\theta$ or $-\Lambda_\theta$ is unital and positive. +\begin{proof} +The first part follows from the above lemma. If $\theta$ is an injection with $\theta(\bT) = \bT$, we have that $S_\theta(\hat{1}) \in \{\hat{1},\widehat{-1}\}$ and that $\Lambda_\theta$ is contractive, so one of $\Lambda_\theta$ or $-\Lambda_\theta$ is a unital contraction, hence positive by part (1) of Proposition \ref{prop:abelian-operator-systems}. +\end{proof} +\end{cor} + +\begin{theorem}\label{thm:affine-map-from-injection} +Suppose $A,B$ are unital C*-algebras and $\theta: U^0(A) \to U^0(B)$ is a contractive injective homomorphism such that $\theta(\bT) = \bT$. Suppose that $T(A) \neq \emptyset$. Then there is a continuous affine map $T_\theta: T(B) \to T(A)$ such that $\Lambda_\theta(f) = f \circ T_\theta$ or $-\Lambda_\theta(f) = f \circ T_\theta$, depending on whether $\Lambda_\theta$ or $-\Lambda_\theta$ is positive. +\begin{proof} +This follows from the fact that the induced map $\Lambda_\theta: \Aff T(A) \to \Aff T(B)$ will have the property that $\Lambda_\theta$ or $-\Lambda_\theta$ will be a unital positive map. Therefore by contravariant identification of compact convex sets (of locally convex Hausdorff linear spaces) with the state space of the space of continuous real-valued affine valued functions on them (\cite[Chapter 7]{Goodearlbook}), there exists a continuous affine map $T_\theta: T(B) \to T(A)$. +\end{proof} +\end{theorem} + +\begin{theorem}\label{thm:trace-result} +Suppose $A,B$ be unital C*-algebras and $\theta: U^0(A) \to U^0(B)$ is a contractive topological group isomorphism such that $\theta(\bT) = \bT$. Suppose that $T(A) \neq \emptyset$. Then the map $T_\theta: T(B) \to T(A)$ induced by $\Lambda_\theta$ is an affine homeomorphism. +\begin{proof} +As $\theta(\bT) = \bT$, $S_\theta(1) \in \{-1,1\}$. Let $\pm \Lambda_\theta: \Aff T(A) \to \Aff T(B)$ be either $\Lambda_\theta$ or $-\Lambda_\theta$, depending on which is unital, positive, contractive and surjective by combining Lemmas \ref{lem:lipschitzSbounded}, \ref{lem:tildeS-isometric} and \ref{lem:sa-bijective}(2). By the duality of (compact) simplices and continuous affine functions on them, the map $T_\theta: T(B) \to T(A)$ is an affine homeomorphism. +\end{proof} +\end{theorem} + +\begin{theorem}\label{thm:KTu-morphism-from-injection} +Let $A,B$ be unital C*-algebras and $\theta: U(A) \to U(B)$ be a contractive injective homomorphism such that $\theta(\bT) = \bT$. Suppose that $T(A) \neq \emptyset$. If +\begin{equation}\label{eq:k-regular-KTu-morph} +\pi_i(U(A)) \simeq K_{i-1}(A)\text{ and } \pi_i(U(B)) \simeq K_{i-1}(B) \text{ for } i=0,1,\footnote{Note that $\pi_1(U(A)) = \pi_1(U^0(A))$ and $\pi_1(U(B)) = \pi_1(U^0(B))$ since we are taking our base point to be the identity in each case.} +\end{equation} +via the canonical maps, then there is an induced map +\begin{equation} +KT_u(\theta): KT_u(A) \to \KT_u(B). +\end{equation} +\begin{proof} +Let +\begin{itemize} +\item $\Lambda:= \Lambda_{\theta|_{U^0(A)}}: \Aff T(A) \to \Aff T(B)$, +\item $\theta_0: \pi_1(U^0(A)) \to \pi_1(U^0(B))$ be the map induced on fundamental groups by $\theta|_{U^0(A)}$, +\item $K_0(\theta)$ be the map induced on $K_0$ by $\theta_0$ together with (\ref{eq:k-regular-KTu-morph}) for $i = 1$, +\item $\theta_1: \pi_0(U(A)) \to \pi_0(U(B))$ be the map induced by $\theta$ on connected components (so that $\theta_1([u]_{\sim_h}) = [\theta_1(u)]_{\sim_h}$) and +\item $K_1(\theta)$ be the map induced by $\theta_1$ together with (\ref{eq:k-regular-KTu-morph}) for $i=0$. +\end{itemize} +Then +\begin{equation} +(\pm K_0(\theta),K_1(\theta),\pm \Lambda): KT_u(A) \to KT_u(B) +\end{equation} +is a $KT_u$-morphism, where $\pm \Lambda$ is either $\Lambda$ or $-\Lambda$ depending on which one is unital and positive, and $\pm K_0(\theta)$ is either $K_0(\theta)$ if $\Lambda$ is positive or $-K_0(\theta)$ if $-\Lambda$ is positive. Indeed, $\pm K_0(\theta),\theta_1,\pm \Lambda$ are all appropriate morphisms, and we have that +\begin{equation} +\begin{tikzcd} +K_0(A) \arrow[d, "\pm K_0(\theta)"'] \arrow[r, "\rho_A"] & \Aff T(A) \arrow[d, "\pm \Lambda"] \\ +K_0(B) \arrow[r, "\rho_B"'] & \Aff T(B) +\end{tikzcd} +\end{equation} +commutes\footnote{The map $-K_0(\theta)$ will take a piece-wise smooth loop $\xi$ to the loop $-\theta \circ \xi$ defined by $(-\theta \circ \xi)(t) = \theta(\xi(-t))$. From here its obvious that the diagram commutes.} by Corollary \ref{cor:pairing}. +\end{proof} +\end{theorem} + +\begin{cor}\label{cor:KTu-isomorphism} +Let $A,B$ be unital C*-algebras, $\theta: U^0(A) \to U^0(B)$ is a contractive topological group isomorphism such that $\theta(\bT) = \bT$, and suppose that $T(A) \neq \emptyset$. If +\begin{equation} +\pi_i(U(A)) \simeq K_{i-1}(A)\text{ and } \pi_i(U(B)) \simeq K_{i-1}(B) \text{ for } i=0,1, +\end{equation} +via the canonical maps, then $KT_u(A) \simeq KT_u(B)$. +\begin{proof} +By Corollary \ref{thm:KTu-morphism-from-injection}, we have an induced $KT_u$-morphism. This map is necessarily an isomorphism since $\theta$ is. +\end{proof} +\end{cor} + +\begin{cor}\label{cor:reg-KT_u-morph} +Let $A,B$ be unital C*-algebras which are either $\cZ$-stable or of stable rank one and suppose that $T(A) \neq \emptyset$. Let $\theta: U(A) \to U(B)$ be a contractive injective homomorphism such that $\theta(\bT) = \bT$. Then there is an induced map +\[ KT_u(\theta): KT_u(A) \to KT_u(B). \] +\begin{proof} +C*-algebras which are $\cZ$-stable or have stable rank one satisfy the hypotheses of Theorem \ref{thm:KTu-morphism-from-injection} by \cite{Jiang97} and \cite{Rieffel87} respectively. So Theorem \ref{thm:KTu-morphism-from-injection} applies. +\end{proof} +\end{cor} + +\begin{comment} +\begin{cor} +Let $A,B$ be unital, separable, stable finite, $\cZ$-stable C*-algebras. Assume that $B$ is simple and let $\theta: U(A) \to U(B)$ be a contractive topological group isomorphism. Then $KT_u(A) \simeq KT_u(B)$. +\begin{proof} +By \cite{Jiang97}, $A,B$ satisfy condition 1 of \ref{thm:same-KTu}, and condition two follows +\end{proof} +\end{cor} +\end{comment} + +\begin{remark}\label{rem:strict-order} +The strict ordering on $\Aff T(A)$ is given by $f \gg g$ if $f(\tau) > g(\tau)$ for all $\tau \in T(A)$. If $A,B$ are unital with $T(A) \neq \emptyset$ and $\theta:U^0(A) \to U^0(B)$ is a contractive injective homomorphism such that $\theta(\bT) = \bT$, then $\pm \Lambda_\theta: \Aff T(A) \to \Aff T(B)$ is a unital positive contraction by Lemma \ref{lem:tildeS-isometric} (again $\pm \Lambda_\theta$ is $\Lambda_\theta$ or $-\Lambda_\theta$ depending on which is positive). We moreover have that +\begin{equation} +\pm \Lambda_\theta(f) \gg \pm \Lambda_\theta(g) \iff f \gg g. +\end{equation} +Indeed, let us show that $f \gg 0$ if and only if its image is $\gg 0$. As $\pm \Lambda_\theta$ has the form $\pm \Lambda_\theta(\hat{a}) = \widehat{\pm S_\theta(a)}$, it suffices to show that if $\sigma(a) > 0$ for all $\sigma \in T(A)$, then $\tau(\pm S_\theta (a)) > 0$ for all $\tau \in T(B)$. +But this is trivial because $\tau \circ \pm S_{\theta}: A_{sa} \to \bR$ extends canonically to a tracial state $A \to \bC$ by Proposition \ref{prop:induced-aff-map}, so evaluating it against $a$ must give that it is strictly positive. +\end{remark} + +The above says the following: for certain C*-algebras, we can read off positivity in $K_0$, thinking of it as the fundamental group of the unitary group, from the strict positivity of the pre-determinant applied a the loop. Precisely, a non-zero element $x \in K_0(A)$, where $A$ is a unital, simple C*-algebra with strict comparison, is in the positive cone if and only if the corresponding loop $\xi_x$ satisfies $\tD_\tau(\xi_x) > 0$ for all $\tau \in T(A)$. + +Although the following is known, for example by very strong results in \cite[Chapter 6]{AraMathieubook} pertaining to certain prime C*-algebras, we give the follow\hyp{}ing as a corollary by using $K$-theoretic classification results. + +\begin{cor} +Let $A,B$ be unital, separable, simple, nuclear $\cZ$-stable C*-algebras satisfying the UCT. Then $A \simeq B$ if and only if there is a contractive isomorphism $U(A) \simeq U(B)$. +\begin{proof} +Its clear that two isomorphic C*-algebras have isomorphic unitary groups. On the other hand, if $U(A) \simeq U(B)$, then since these C*-algebras are $\cZ$-stable, Corollary \ref{cor:KTu-isomorphism} applies. As $KT_u(\cdot)$ recovers the Elliott invariant, which is a complete invariant for the C*-algebras as in the statement of the theorem (by \cite[Corollary D]{CETWW21}, \cite{EGLN15,GongLinNiu20I,GongLinNiu20II} and the references therein), $A \simeq B$. +\end{proof} +\end{cor} + +Using the state of the art classification of embeddings \cite{CGSTW23}, there is an enlarged invariant of $KT_u(\cdot)$ which is able to classify morphisms between certain C*-algebras. Any $KT_u$-morphisms automatically has a lift to this larger invariant \cite[Theorem 3.9]{CGSTW23}, and so under the assumption that the $KT_u$-morphism is faithful (i.e., the map $T(B) \to T(A)$ induced by the map $\Aff T(A) \to \Aff T(B)$ sends traces on $B$ to faithful traces on $A$), there is a *-homomorphism witnessing the $KT_u$-morphism. Therefore as a corollary of their main theorem, we have that for an abundance of C*-algebras, there is an (contractive) embedding of unitary groups if and only if there is an embedding of C*-algebras. + +\begin{cor}\label{cor:ugh-subgroup-subalgebra} +Let $A$ be a unital, separable, simple nuclear C*-algebra sat\hyp{}isfying the UCT which is either $\cZ$-stable or of stable rank one, and $B$ be a unital, separable, simple, nuclear $\cZ$-stable C*-algebra. If there is a contractive injective homomorphism $\theta:U(A) \to U(B)$ such that $\theta(\bT) = \bT$, then there is an embedding $A \into B$. +\begin{proof} +Assuming such a $\theta$ exists, it gives rise to a $KT_u$-morphism +\begin{equation} +KT_u(\theta): KT_u(A) \to KT_u(B) +\end{equation} +by Corollary \ref{cor:reg-KT_u-morph}. As $A,B$ are simple, the map $T_\theta: T(B) \to T(A)$ necessarily maps traces on $B$ to faithful traces on $A$, and thus the $KT_u$-morphism $KT_u(\theta)$ is ``faithful''. Therefore $KT_u(\theta)$ induces an embedding $A \into B$ by \cite[Theorem B]{CGSTW23}. +\end{proof} +\end{cor} + + +\section{A slight general linear variant}\label{section:ugh-general-linear-variants} + +Here we briefly describe a slight general linear variant of some of the results above. +Unfortunately, the maps we get at the level of $A,B$ and complex-valued continuous affine functions are not $\bC$-linear in general (see Example \ref{example:non-C-linear}). +In the presence of a continuous homomorphism $\theta: GL^0(A) \to GL^0(B)$, we have corresponding maps +\begin{equation}\label{eq:gch-commute} +\begin{tikzcd} +GL^0(A)/CGL^0(A) \arrow[r, "\simeq"] \arrow[d] & \left(A/\ov{[A,A]}\right)/\ov{\tD_A^1(\pi_1(U^0(A)))} \arrow[d] \\ +GL^0(B)/CGL^0(B) \arrow[r, "\simeq"'] & \left(A/\ov{[A,A]}\right)/\ov{\tD_B^1(\pi_1(U^0(B)))} . +\end{tikzcd} +\end{equation} +Again, by modding out by algebraic commutator subgroups when $DGL^0(A) = \ker \Delta_A^1$ and $DGL^0(B) = \ker \Delta_B^1$ (both with respect to the general linear variant of the de la Harpe-Skandalis determinant, as originally introduced in \cite{dlHS84a}), instead of closures of derived groups, there is a purely algebraic variant of the above diagram: +\begin{equation}\label{eq:gh-commute} +\begin{tikzcd} +GL^0(A)/DGL^0(A) \arrow[r, "\simeq"] \arrow[d] &\left(A/\ov{[A,A]}\right)/\tD_A^1(\pi_1(GL^0(A))) \arrow[d] \\ +GL^0(B)/DGL^0(B) \arrow[r, "\simeq"'] & \left(A/\ov{[B,B]}\right)/\tD_B^1(\pi_1(GL^0(B))). +\end{tikzcd} +\end{equation} + +Thinking of $K_0(A)$ as the Grothendieck group of the semigroup of equivalence classes of idempotents and $K_0(A) \simeq \pi_1(GL_{\infty}^0(A))$, we would like to lift the maps on the right of (\ref{eq:gch-commute}) and (\ref{eq:gh-commute}) +to a map +\begin{equation} +A/\ov{[A,A]} \to B/\ov{[B,B]} +\end{equation} +(the latter holding true when $A,B$ are C*-algebras whose determinant has appropriate kernel). + +We can always achieve a bounded $\bR$-linear map. + +\begin{prop}\label{prop:gl-G-theta-lift} +Let $A,B$ be unital C*-algebras and $\theta: GL^0(A) \to GL^0(B)$ be a continuous group homomorphism. Then there is a continuous $\bR$-linear map +\begin{equation} +\tilde{G}_\theta: A/\ov{[A,A]} \to B/\ov{[B,B]} +\end{equation} +which lifts the maps on the right of (\ref{eq:gch-commute}) and (\ref{eq:gh-commute}) (the latter holding when $DGL^0(A) = \ker\Delta_A^1$ and $DGL^0(B) = \ker\Delta_B^1$). +\begin{proof} + + + +\begin{comment} +Since $\theta$ sends unitaries to unitaries, let $S_\theta$ be as in Section \ref{section:ugh-ctugh-and-traces} and define +\begin{equation} +G_\theta(a) := S_\theta(\text{Re}(a)) + iS_\theta(\text{Im}(a)). +\end{equation} +This map is clearly continuous and linear. +, and it is linear since if $s + ti \in \bC$ and $a,b \in A$, we have +\begin{equation} +\begin{split} +G_\theta((s+ti)a + b) &= S_\theta(\text{Re}((s + ti)a + b)) + iS_\theta(\text{Im}((s + ti)a + b) \\ +&= S_\theta\left(s\text{Re}(a) - t\text{Im}(a) + \text{Re}(b)\right) \\ +&\ \ \ \ \ +iS_\theta\left(s\text{Im}(a) + t\text{Re}(a) + \text{Im}(b)\right) \\ +&= sS_\theta(\text{Re}(a)) -tS_\theta(\text{Im}(a)) + S_\theta(\text{Re}(b)) \\ +&\ \ \ \ \ + isS_\theta(\text{Im}(a)) + itS_\theta(\text{Re}(a)) + iS_\theta(\text{Im}(b)) \\ +&= (s + it)S_\theta(\text{Re}(a)) + i(s+it)S_\theta(\text{Im}(a) + G_\theta(b) \\ +&=(s + it)G_\theta(a) + G_\theta(b). +\end{split} +\end{equation} +Now we note that $A_0 + iA_0 = \ov{[A,A]}$, and similarly for $B$. Indeed, inclusion $A_0 + iA_0 \subseteq \ov{[A,A]}$ is clear, identifying $A_0$ with the space of all self-adjoints which vanish on every tracial state, and identifying $\ov{[A,A]}$ with the space of all elements which vanish on every tracial state. For the reverse, we note that +\begin{equation} +\begin{split} +\text{Re}([a,b]) &= \frac{[a,b]}{2} + \frac{[b^*,a^*]}{2} \text{ and }\\ +\text{Im}([a,b]) &= \frac{[a,b]}{2i} - \frac{[b^*,a^*]}{2i} +\end{split} +\end{equation} +are both elements which vanish on every tracial state. Thus$[A,A] \subseteq A_0 + iA_0$, which is closed and we therefore have that $\ov{[A,A]} = A_0 + iA_0$. We can then conclude that $G_\theta(\ov{[A,A]}) \subseteq \ov{[B,B]}$ since $S_\theta(A_0) \subseteq B_0$ by Proposition \ref{prop:induced-aff-map}. + +We now define $\tilde{G}_\theta: A/\ov{[A,A]} \to B/\ov{[A,A]}$ by $\tilde{G}_\theta([a]) = [G_\theta(a)]$. The fact that $\tilde{G}_\theta$ is a lift of the maps on the right of (\ref{eq:gch-commute}) and (\ref{eq:gh-commute}) follows from essentially the same proof as Proposition \ref{prop:S-theta-is-a-lift}. +\end{comment} +We define $G_\theta: A \to B$ given by +\begin{equation} +G_\theta(a) := \lim_n \frac{n}{2\pi i}\log \theta(e^{2\pi i \frac{a}{n}}). +\end{equation} +We note that the sequence on the right is eventually constant: choose $N$ large enough such that $n \geq N$ implies that +\begin{equation} +\|\theta(e^{2\pi i \frac{a}{n}}) - 1\| < 1. +\end{equation} +We then have for $n \geq N$, +\begin{equation} +\begin{split} +\frac{n}{2\pi i}\log \theta(e^{2\pi i \frac{a}{n}}) &= \frac{n}{2\pi i}\log \theta(e^{2\pi i \frac{Na}{Nn}}) \\ +&= \frac{n}{2\pi i}\log \theta(e^{2\pi i \frac{a}{Nn}})^N \\ +&= \frac{Nn}{2\pi i}\log \theta(e^{2\pi i \frac{a}{Nn}}) \\ +&= \frac{N}{2\pi i}\log \theta(e^{2\pi i \frac{a}{Nn}})^n \\ +&= \frac{N}{2\pi i}\log \theta(e^{2\pi i \frac{na}{Nn}}) \\ +&= \frac{N}{2\pi i}\log \theta(e^{2\pi i \frac{a}{N}}). +\end{split} +\end{equation} +To see that the map is additive, one can use the Lie product formula +\begin{equation} +e^{a+b} = \lim_k \left(e^{\frac{a}{k}}e^{\frac{b}{k}}\right)^k +\end{equation} + (see for example \cite[Theorem 2.11]{HallBook}), along with the fact that $\theta$ is a continuous homomorphism. +From here, it is clear that $G_\theta$ is continuous and $\bQ$-linear, hence $\bR$-linear. Moreover, one can use the formula +\begin{equation} +e^{[a,b]} = \lim_k \left(e^{\frac{-a}{k}}e^{\frac{-b}{k}}e^{\frac{a}{k}}e^{\frac{b}{k}}\right)^{k^2} +\end{equation} +(a variation of the argument given in the proof of \cite[Theorem 2.11]{HallBook} will give the desired formula), together with $\theta$ being a continuous group homomorphism, to show that it respects commutation. Note that the same proof indeed shows that $S_\theta$ respects commutation, although we never explicitly used this. +From here, $G_\theta([A,A]) \subseteq [B,B]$, and consequently $G_\theta(\ov{[A,A]}) \subseteq \ov{[B,B]}$ by continuity. +Thus there is an induced $\bR$-linear map +\begin{equation} +\tilde{G}_\theta: A/\ov{[A,A]} \to B/\ov{[B,B]}. +\end{equation} +The fact that $\tilde{G}_\theta$ is a lift of the maps on the right of (\ref{eq:gch-commute}) and (\ref{eq:gh-commute}) follows from the same arguments as in Propositions \ref{prop:S-theta-is-a-lift} and \ref{prop:S-theta-is-a-lift-H}. +\end{proof} +\end{prop} + +\begin{remark} +We note that $G_\theta$ could have also been defined in the same manner as $S_\theta$. In particular, there is a correspondence +\begin{equation}\label{eq:G-theta-correspondence} +\theta(e^{2\pi i t a}) = e^{2\pi i t G_\theta(a)}, t \in \bR. +\end{equation} +One can use this to show that $\theta(\ker \Delta_A^1) \subseteq \ker \Delta_B^1$ as in Proposition \ref{prop:pre-det-commuting-diagrams} by using $G_\theta$ in place of $S_\theta$, along with the fact that $G_\theta(\ov{[A,A]}) \subseteq \ov{[B,B]}$. So there will always be a general linear variant of the commuting diagram (\ref{eq:h-commute-pre-det}). +\end{remark} + +Let us use (\ref{eq:G-theta-correspondence}) to show that the maps $G_\theta$ and $\tilde{G}_\theta$ in Proposition \ref{prop:gl-G-theta-lift} are not always $\bC$-linear. + +\begin{example}\label{example:non-C-linear} +Consider $A = B = \bC$ and $\theta: \bC^\times \to \bC^\times$ given by $\theta(z) = |z|^{(\alpha + \beta i)}z^n$ where $\alpha,\beta \in \bR$ and $n \in \bZ$. It is easy to see that $\theta$ is a continuous group homomorphism. However, the map $G_\theta$ is not $\bC$-linear. Indeed, we have that +\begin{equation} +\begin{split} +\theta(e^{2\pi i t(a+bi)}) &= |e^{2\pi i t(a + bi)}|^{\alpha + \beta i}e^{2\pi i t n(a + bi)} \\ +&= (e^{-2\pi t b})^{\alpha + \beta i}e^{2\pi i t n(a + bi)} \\ +&= e^{-2\pi t b\alpha}e^{-2\pi i t\beta i b}e^{2\pi i t n(a + bi)} \\ +&= e^{2\pi i t((na - \alpha b) + i(n - \beta)b}. +\end{split} +\end{equation} +In particular, thinking of $\bC$ as $\bR^2$ with 1 and $i$ corresponding to the basis vectors $(1,0)$ and $(0,1)$ respectively, we have that $G_\theta: \bR^2 \to \bR^2$ is the map +\begin{equation} +G_\theta\begin{pmatrix} +a \\ b +\end{pmatrix} = \begin{pmatrix} +n & -\alpha \\ 0 & n - \beta +\end{pmatrix}\begin{pmatrix} +a \\ b +\end{pmatrix}. +\end{equation} +In this example, the map $G_\theta$ is $\bC$-linear if and only if $\alpha = \beta = 0$. We do note, however, that $\theta$ sends unitaries to unitaries and $\theta|_{\bT}(z) = z^n$. +\end{example} + +In general, it is clear that if $\theta: GL(A) \to GL(B)$ sends unitaries to unitaries, then we can use techniques in Section \ref{section:ugh-order-on-aff-classification} to get maps between spaces of continuous affine functions on the trace simplices. If one had that $\theta$ was the restriction of a *-homomorphism or a conjugate-linear *-homomorphism, then this would be true. + +\section{Final remarks and open questions}\label{section:ugh-final-remarks} + +An alternate way to construct the map $\Lambda_\theta$, using duality of traces, is as follows. Denote by $\fT_s(A)$ the set of all tracial functionals on $A$. Suppose that $A,B$ are unital C*-algebras with $T(A),T(B) \neq \emptyset$ and $\theta: U^0(A) \to U^0(B)$ is a continuous homomorphism. Define, for $a \in A_{sa}$ and $\tau \in \fT_s(B)$, +\begin{equation}\label{eq:F-tau-def} +F(\tau)(a) := \lim_{n \to \infty} \tau\left(\frac{n}{2\pi i}\log \theta(e^{2\pi i \frac{a}{n}})\right). +\end{equation} + +\begin{prop} +For $\tau \in \fT_s(B)$, the map $F(\tau): A_{sa} \to \bR$ is a well-defined, bounded, self-adjoint, tracial functional. Moreover, $F: \fT_s(B) \to \fT_s(A)$ given by $\tau \mapsto F(\tau)$ is a bounded $\bR$-linear map. +\end{prop} +\begin{proof} +Using the same arguments as in Proposition \ref{prop:gl-G-theta-lift}, it is clear the formula (\ref{eq:F-tau-def}) is well-defined and gives rise to a bounded $\bR$-linear map $F: \fT_s(B) \to \fT_s(A)$. +\end{proof} + +One can identify $\left(A_{sa}/A_0\right)^* \simeq \fT_s(A)$ (see, for example \cite{CuntzPedersen79}), and so we can use duality to get a map $\tilde{\Lambda}_\theta:= F^*: \fT_s(A)^* \to \fT_s(B)^*$ and restrict it to the dense set $A_{sa}/A_0$. One can check that the image lies in $B_{sa}/B_0$ and that the restriction is just the map $\Lambda_\theta$ that we got before. + +We finish by listing some open problems. + +\begin{enumerate} +\item There are classes where topological isomorphisms between $U(A)$ and $U(B)$ (or even $U^0(A)$ and $U^0(B)$) come from *-isomorphisms or anti-*-isomorphisms. For example, if $A,B$ are prime traceless C*-algebras containing full square zero elements, this is true by results in \cite{ChandRobert23}. + +If $A$ is a unital, separable, nuclear C*-algebra satisfying the UCT and $B$ is a unital simple separable nuclear $\cZ$-stable C*-algebra, then unital embeddings $A \into B$ are classified by an invariant $\uv{K}T_u(\cdot)$ which is an enlargement of $KT_u$ \cite{CGSTW23}. Thus any isometric unitary group homomorphism $U(A) \to U(B)$ will give a $KT_u$-morphism $KT_u(\theta)$ and therefore there will be an embedding $\phi:A \into B$ such that $KT_u(\phi) = KT_u(\theta)$. However it is not clear that $\phi$ satisfies $\phi|_{U(A)} = \theta$. More generally though -- in the tracial setting -- are there continuous group homomorphisms which do not have lifts to *-homomorphisms or anti-*-homomorphisms? + +Note that in \cite[Chapter 6]{AraMathieubook}, Lie isomorphisms between certain C*-algebras are shown to be the sum of a Jordan *-isomorphism and a center-valued trace. Is there a result for general (injective) Lie homomorphisms between certain classes of C*-algebras? + +\item This enlargement of $KT_u$ discussed in \cite{CGSTW23} contains $K$-theory with coefficients (along with appropriate pairing maps -- the Bosckstein maps discussed in \cite{SchochetIV}). So we ask: do continuous group homo\hyp{}morphisms induce maps between $K$-theory with coefficients? + +\item For a general continuous homomorphism $\theta: U^0(A) \to U^0(B)$, does the norm $\|S_\theta\|$ determine a Lipschitz constant for $\theta$? We clearly have that +\begin{equation} +\|S_\theta\| \leq \inf\{ K \mid \theta \text{ is }K\text{-Lipschitz}\} +\end{equation} +by Lemma \ref{lem:lipschitzSbounded}. Is this equality? +\item For $A$ simple (or prime), is it true that any continuous injective homo\hyp{}morphism $\theta: U^0(A) \to U^0(B)$ is isometric? Contractive? What if $B$ is simple (or prime)? + +\item In the initial draft of this paper, we claimed that any continuous group homomorphism in Proposition \ref{prop:gl-G-theta-lift} gave rise to a $\bC$-linear $\tilde{G}_\theta$. This is clearly false by Example \ref{example:non-C-linear}. Is there a way to guarantee that the map $\tilde{G}_\theta$ is $\bC$-linear? Or can one alter it accordingly for this to happen? Or alter it to get a map between unitary groups, which would then allow one to use the results in Section \ref{section:ugh-order-on-aff-classification}? Maybe if one starts with an injective, contractive group homomorphism $GL^0(A) \to GL^0(B)$ which sends $\bC^\times$ to $\bC^\times$, one can say something. +\end{enumerate} + + + + \bibliographystyle{amsalpha} + \bibliography{biblio} + +\end{document} diff --git a/Unitary groups, K-theory and traces/unitary_group_homs.toc b/Unitary groups, K-theory and traces/unitary_group_homs.toc new file mode 100644 index 0000000..4c5831c --- /dev/null +++ b/Unitary groups, K-theory and traces/unitary_group_homs.toc @@ -0,0 +1,15 @@ +\babel@toc {nil}{}\relax +\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{section.1}% +\setcounter {tocdepth}{0} +\contentsline {section}{\tocsection {}{}{Acknowledgements}}{4}{section*.2}% +\setcounter {tocdepth}{1} +\contentsline {section}{\tocsection {}{2}{Preliminaries and notation}}{4}{section.2}% +\contentsline {subsection}{\tocsubsection {}{2.1}{Notation}}{4}{subsection.2.1}% +\contentsline {subsection}{\tocsubsection {}{2.2}{The de la Harpe--Skandalis determinant and Thomsen's variant}}{5}{subsection.2.2}% +\contentsline {subsection}{\tocsubsection {}{2.3}{Thomsen's variant}}{8}{subsection.2.3}% +\contentsline {subsection}{\tocsubsection {}{2.4}{The $KT_u$-invariant}}{9}{subsection.2.4}% +\contentsline {section}{\tocsection {}{3}{Continuous unitary group homomorphisms and traces}}{9}{section.3}% +\contentsline {section}{\tocsection {}{4}{The order on $\Aff T(\cdot )$}}{19}{section.4}% +\contentsline {section}{\tocsection {}{5}{A slight general linear variant}}{25}{section.5}% +\contentsline {section}{\tocsection {}{6}{Final remarks and open questions}}{28}{section.6}% +\contentsline {section}{\tocsection {}{}{References}}{29}{section*.3}% diff --git a/Universal covering groups of unitary groups of von Neumann algebras/Universal_cover_of_U_M.bbl b/Universal covering groups of unitary groups of von Neumann algebras/Universal_cover_of_U_M.bbl new file mode 100644 index 0000000..da7bf5e --- /dev/null +++ b/Universal covering groups of unitary groups of von Neumann algebras/Universal_cover_of_U_M.bbl @@ -0,0 +1,127 @@ +\newcommand{\etalchar}[1]{$^{#1}$} +\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} +\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } +% \MRhref is called by the amsart/book/proc definition of \MR. +\providecommand{\MRhref}[2]{% + \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} +} +\providecommand{\href}[2]{#2} +\begin{thebibliography}{CGS{\etalchar{+}}23} + +\bibitem[AS00]{AguilarSocolovsky00} +Marcelo~A. Aguilar and Miguel Socolovsky, \emph{Universal covering group of + {${\rm U}(n)$} and projective representations}, Internat. J. Theoret. Phys. + \textbf{39} (2000), no.~4, 997--1013. \MR{1779169} + +\bibitem[ASS71]{ArakiSmithSmith71} +Huzihiro Araki, Mi-Soo~B. Smith, and Larry Smith, \emph{On the homotopical + significance of the type of von {N}eumann algebra factors}, Comm. Math. Phys. + \textbf{22} (1971), 71--88. \MR{288587} + +\bibitem[Bre70]{Breuer70} +M.~Breuer, \emph{On the homotopy type of the group of regular elements of + semifinite von {N}eumann algebras}, Math. Ann. \textbf{185} (1970), 61--74. + \MR{264408} + +\bibitem[Bro67]{Broise67} +Michel Broise, \emph{Commutateurs dans le groupe unitaire d'un facteur}, J. + Math. Pures Appl. (9) \textbf{46} (1967), 299--312. \MR{223900} + +\bibitem[BW76]{BruningWillgerodt76} +Jochen Br\"uning and Wolfgang Willgerodt, \emph{Eine {V}erallgemeinerung eines + {S}atzes von {N}. {K}uiper}, Math. Ann. \textbf{220} (1976), no.~1, 47--58. + \MR{405483} + +\bibitem[CGS{\etalchar{+}}23]{CGSTW23} +Jos{\'e}~R. Carri{\'o}n, James Gabe, Christopher Schafhauser, Aaron Tikuisis, + and Stuart White, \emph{Classifying *-homomorphisms {I}: unital simple + nuclear {C}*-algebras}, arXiv:2307.06480 (2023). + +\bibitem[dlH13]{dlHarpe13} +Pierre de~la Harpe, \emph{Fuglede--{K}adison determinant: theme and + variations}, Proc. Natl. Acad. Sci. USA \textbf{110} (2013), no.~40, + 15864--15877. \MR{3363445} + +\bibitem[dlHM83]{dlHarpeMcDuff83} +Pierre de~la Harpe and Dusa McDuff, \emph{Acyclic groups of automorphisms}, + Comment. Math. Helv. \textbf{58} (1983), no.~1, 48--71. \MR{699006} + +\bibitem[dlHS84]{dlHS84a} +Pierre de~la Harpe and Georges Skandalis, \emph{D\'{e}terminant associ\'{e} \`a + une trace sur une alg\'{e}bre de {B}anach}, Ann. Inst. Fourier (Grenoble) + \textbf{34} (1984), no.~1, 241--260. \MR{743629} + +\bibitem[FdlH80]{FackdlHarpe80} +Thierry Fack and Pierre de~la Harpe, \emph{Sommes de commutateurs dans les + alg\`ebres de von {N}eumann finies continues}, Ann. Inst. Fourier (Grenoble) + \textbf{30} (1980), no.~3, 49--73. \MR{597017} + +\bibitem[FK52]{FugledeKadison52} +Bent Fuglede and Richard~V. Kadison, \emph{Determinant theory in finite + factors}, Ann. of Math. (2) \textbf{55} (1952), 520--530. \MR{52696} + +\bibitem[Han78]{Handelman78} +David Handelman, \emph{{$K_0$} of von {N}eumann and {AF} {C}*-algebras}, Quart. + J. Math. Oxford Ser. (2) \textbf{29} (1978), no.~116, 427--441. \MR{517736} + +\bibitem[Hat02]{HatcherAT} +Allen Hatcher, \emph{Algebraic topology}, Cambridge University Press, + Cambridge, 2002. \MR{1867354} + +\bibitem[JS99]{JiangSu99} +Xinhui Jiang and Hongbing Su, \emph{On a simple unital projectionless + {C}*-algebra}, Amer. J. Math. \textbf{121} (1999), no.~2, 359--413. + \MR{1680321} + +\bibitem[Ker70]{Kervaire70} +Michel~A. Kervaire, \emph{Multiplicateurs de schur et k-th{\'e}orie}, + pp.~212--225, Springer Berlin Heidelberg, Berlin, Heidelberg, 1970. + +\bibitem[KR86]{KadisonRingroseII} +Richard~V. Kadison and John~R. Ringrose, \emph{Fundamentals of the theory of + operator algebras. {V}ol. {II}}, Pure and Applied Mathematics, vol. 100, + Academic Press, Inc., Orlando, FL, 1986, Advanced theory. \MR{859186} + +\bibitem[Kui65]{Kuiper65} +Nicolaas~H. Kuiper, \emph{The homotopy type of the unitary group of {H}ilbert + space}, Topology \textbf{3} (1965), 19--30. \MR{179792} + +\bibitem[NT96]{NielsenThomsen96} +Karen~E. Nielsen and Klaus Thomsen, \emph{Limits of circle algebras}, + Exposition. Math. \textbf{14} (1996), no.~1, 17--56. \MR{1382013} + +\bibitem[Rie87]{Rieffel87} +Marc~A. Rieffel, \emph{The homotopy groups of the unitary groups of + noncommutative tori}, J. Operator Theory \textbf{17} (1987), no.~2, 237--254. + \MR{887221} + +\bibitem[RLL00]{RordamKBook} +Mikael R{\o}rdam, Flemming Larsen, and Niels~J. Laustsen, \emph{An introduction + to {K}-theory for {C}*-algebras}, London Mathematical Society Student Texts, + vol.~49, Cambridge University Press, Cambridge, 2000. \MR{1783408} + +\bibitem[Rot88]{RotmanATBook} +Joseph~J. Rotman, \emph{An introduction to algebraic topology}, Graduate Texts + in Mathematics, vol. 119, Springer-Verlag, New York, 1988. \MR{957919} + +\bibitem[Sch84a]{Schroder84real} +Herbert Schr{\"o}der, \emph{On the homotopy type of the regular group of a real + {W}*-algebra}, Integral Equations and Operator Theory \textbf{9} (1984), + 694--705. + +\bibitem[Sch84b]{Schroder84} +\bysame, \emph{On the homotopy type of the regular group of a {W}*-algebra.}, + Mathematische Annalen \textbf{267} (1984), 271--278. + +\bibitem[Tak02]{TakesakiI} +Masamichi Takesaki, \emph{Theory of operator algebras. {I}}, Encyclopaedia of + Mathematical Sciences, vol. 124, Springer-Verlag, Berlin, 2002, Reprint of + the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5. + \MR{1873025} + +\bibitem[Tho95]{Thomsen95} +Klaus Thomsen, \emph{Traces, unitary characters and crossed products by + {$\mathbb{Z}$}}, Publ. Res. Inst. Math. Sci. \textbf{31} (1995), no.~6, + 1011--1029. \MR{1382564} + +\end{thebibliography} diff --git a/Universal covering groups of unitary groups of von Neumann algebras/Universal_cover_of_U_M.tex b/Universal covering groups of unitary groups of von Neumann algebras/Universal_cover_of_U_M.tex new file mode 100644 index 0000000..cf9a4e3 --- /dev/null +++ b/Universal covering groups of unitary groups of von Neumann algebras/Universal_cover_of_U_M.tex @@ -0,0 +1,415 @@ +%\documentclass[12pt]{amsart} +\documentclass{amsart} + +%\usepackage[margin=1.5in]{geometry} + +\include{preabmle} +\include{macros} +\include{letterfonts} + +\begin{document} +\title[Universal covering groups of unitary groups of W*-algebras]{Universal covering groups of unitary groups of von Neumann algebras} + +%Pawel 0000-0002-6941-264X +\author[Pawel Sarkowicz]{Pawel Sarkowicz} +\address[Pawel Sarkowicz]{Department of Pure Mathematics, University of Waterloo, N2L 3G1, Canada} +\email{\href{mailto:psarkowi@uwaterloo.ca}{psarkowi@uwaterloo.ca}} + + \begin{abstract} + We give a short and simple proof, utilizing the pre-determinant of P. de la Harpe and G. Skandalis, that the universal covering group of the unitary group of a $\II_1$ von Neumann algebra $\cM$, when equipped with the norm topology, splits algebraically as the direct product of the self-adjoint part of its center and the unitary group $U(\cM)$. Thus, when $\cM$ is a $\II_1$ factor, the universal covering group of $U(\cM)$ is algebraically isomorphic to the direct product $\bR \times U(\cM)$. + In particular, the question of P. de la Harpe and D. McDuff of whether the universal cover of $U(\cM)$ is a perfect group is answered in the negative. + \end{abstract} + + \maketitle + \tableofcontents + + \section{Introduction} + +The universal covering group of a topological group, recoverable as the space of paths modulo null-homotopy when it is sufficiently connected, gives rise to an extension of the group by its fundamental group. + The universal covering group of $U(n)$, the unitary group of the $n \times n$ matrices, is known to homeomorphically isomorphic to $SU(n) \rtimes \bR$, where $SU(n)$ is the special unitary group consisting of unitaries with trivial determinant. +Indeed, $U(n)$ itself is the semidirect product $SU(n) \rtimes \bT$ as can see the from the topologically split extension +\begin{equation}\label{eq:U(n) splitting} +\begin{tikzcd} +1 \arrow[r] & SU(n) \arrow[r] & U(n) \arrow[r, "\det"] & \bT \arrow[l, bend right=49] \arrow[r] & 1. +\end{tikzcd} +\end{equation} +Therefore $U(n)$ is covered by $SU(n) \times \bT$, which is further covered by $SU(n) \times \bR$, and this must necessarily be the universal cover, as a topological space, since it is simply connected. +The isomorphism of the universal cover, as a topological group, with the semidirect product $SU(n) \rtimes \bR$ was given in \cite{AguilarSocolovsky00}. + + If $\cM$ is an infinite von Neumann factor, then it is known that the universal covering group is in fact just $U(\cM)$ due to the fact that the fundamental group will be trivial -- see \cite{Kuiper65,Breuer70,ArakiSmithSmith71,BruningWillgerodt76,Handelman78,Schroder84,Schroder84real} for results on homotopy groups related to von Neumann algebras. + Therefore to understand the universal covering group of unitary groups of von Neumann factors, it remains to study them in the $\II_1$ setting. + We show that the universal covering group of $U(\cM)$, where $\cM$ is any $\II_1$ von Neumann algebra, exhibits similar and even simpler algebraic behaviour to that of the universal covering group of $U(n)$. + + +\begin{result} +Let $\cM$ be a type $\II_1$ von Neumann algebra. Then the universal covering group of $U(\cM)$ exists and splits algebraically as the direct product +\begin{equation} +\widetilde{U(\cM)} \simeq Z(\cM)_{sa} \times U(\cM). +\end{equation} +In particular, when $\cM$ is a $\II_1$ factor, +\begin{equation} +\widetilde{U(\cM)} \simeq \bR \times U(\cM). +\end{equation} +\end{result} + +We however note that this is not a topological product. Indeed, $\pi_1(U(\cM))$ is viewed as a discrete subset. Even if $\bR$ was equipped with its standard topology, $U(\cM)$ is not simply connected in the $\II_1$ setting, so that the product space would not be. + +One observation is that there is a way to define a determinant map on the group of invertibles of a $\II_1$ factor $\cM$ via the Fuglede-Kadison determinant \cite{FugledeKadison52} or the de la Harpe-Skandalis determinant \cite{dlHS84a} (they are related -- see \cite{dlHarpe13} for exposition). +Restricting this map to the unitary group, it is a feature that this restriction gives the trivial map. +In particular, if one were to define the \emph{special unitary group} of $\cM$ to be those unitaries in $\cM$ which have trivial determinant, we would have that $SU(\cM) = U(\cM)$, and so the splitting above can be viewed as the direct product of $\bR$ with the special unitary group of $\cM$. +In this sense, this theorem above is an algebraic generalization of the fact that the unitary group $U(n)$ has covering group $SU(n) \rtimes \bR$. + +In \cite{dlHarpeMcDuff83}, de la Harpe and McDuff examined the homology groups of certain automorphism groups arising in operator algebras. +Among other things, they showed that the unitary groups of properly infinite von Neumann algebras are acyclic -- that is, all of their integral homology groups vanish. +The unitary group of a $\II_1$ factor $\cM$ has vanishing first integral homology group (since it is perfect -- see \cite{Broise67,FackdlHarpe80}), but it came into question whether or not the second integral homology group of $U(\cM)$ vanishes. +If one could show that the universal covering group $\widetilde{U(\cM)}$ was perfect, then this group would be a cover in the algebraic sense \cite{Kervaire70} and there would be potential for non-triviality of the second homology group. +However, as a consequence of the main theorem, we answer the question about the universal covering being perfect in the negative. + +\begin{resultcor} +Let $\cM$ be a $\II_1$ factor. Then the universal covering group of $U(\cM)$ is not a perfect group. +\end{resultcor} +The question of the whether or not the second integral homology group (or all of the higher integral homology groups) of $U(\cM)$ is trivial, where $\cM$ is a $\II_1$ factor, remains open. + + + %Acknowledgements section + \addtocontents{toc}{\protect\setcounter{tocdepth}{0}} + \section*{Acknowledgements} + Thanks to Mehdi Moradi for insightful conversations, as well as to Aaron Tikuisis and Laurent Marcoux for looking over a first draft. + + %go back to normal TOC ordering. + \addtocontents{toc}{\protect\setcounter{tocdepth}{1}} + +\section{Preliminaries}\label{sec:prelim} +%\subsection*{Preliminaries and notation}\label{subs:notation} +\subsection{Notation} +For a group $G$, we denote by $DG$ the derived subgroup of $G$, i.e., +\begin{equation} +DG := \langle ghg^{-1}h^{-1} \mid g,h \in G \rangle +\end{equation} +is the subgroup generated by commutators. +If $G$ has an underlying topology, we denote by $G^0$ the connected component of the identity, and we write +\begin{equation} +PG= \{\xi: [0,1] \to G \mid \xi \text{ is continuous, } \xi(0) = 1_G \} +\end{equation} +to be the path group. Multiplication is given by the pointwise product and the topology is the compact-open topology, meaning the topology generated by subbase consisting of sets +\begin{equation} +V(K,O) = \{ f \in C([0,1],G) \mid f(K) \subseteq O\}, +\end{equation} +where $K \subseteq [0,1]$ is closed and $O \subseteq G$ is open. +For two paths $\xi,\eta \in PG$, we will write $\xi \sim_{nh} \eta$, if $\xi$ and $\eta$ are \emph{null-homotopic}, meaning that the path $\xi^{-1}\eta$ is homotopic to the constant path at our base point (in this case, the identity of $G$). In particular, two null-homotopic paths must have the same endpoint. + +For a unital C*-algebra $A$, $A_{sa}$ will denote the self-adjoint elements of $A$, while $U(A)$ will denote the group of unitaries in $A$: that is, the subgroup of invertible elements $u \in A$ satisfying +\begin{equation} +u^*u = uu^* = 1_A. +\end{equation} +We give $U(A)$ the subspace topology induced by the norm of $A$. +For $n \in \bN$, we let $U_n(A) = U(M_n(A))$ and take +\begin{equation} +U_{\infty}(A) = \dlim U_n(A) +\end{equation} +to be the direct limit with connecting maps $u \mapsto u \oplus 1$. +We equip $U_{\infty}(A)$ with the norm topology.\footnote{One can also equip $U_{\infty}(A)$ with the inductive limit topology. However, even though $U_n(A)$ are topological groups, $U_{\infty}(A)$ is almost never a topological group because multiplication will not be jointly continuous in general. The quotient of the closure of the derived group can be more useful (for example, in terms of classification of morphisms) when one uses the inductive limit topology as opposed to the norm topology -- see \cite{Thomsen95,NielsenThomsen96,CGSTW23}.} +The topological $K_1$ group of a unital C*-algebra $A$ is then the set of connected components of $U_{\infty}(A)$ +\begin{equation} +K_1(A) := U_{\infty}(A)/U_{\infty}^0(A). +\end{equation} +For two projections $p,q$ over matrix algebras over $A$, we write $p \MvN q$ if they are Murray-von Neumann equivalent. +The $K_0$ group of a unital C*-algebra is the Grothendieck group of the Murray-von Neumann semigroup of equivalence classes of projections in matrix algebras over $A$, and there is a canonical isomorphism +\begin{equation} +K_0(A) \simeq \pi_1(U_{\infty}(A)) +\end{equation} +which is induced by the map which takes a projection $p \in M_n(A)$ to the loop +\begin{equation} +t \mapsto e^{2\pi i t}p + 1-p +\end{equation} +in $U_n(A) \subseteq U_{\infty}(A)$. +Thus we have +\begin{equation} +K_n(A) = \begin{cases} +\pi_1(U_{\infty}(A)) & n =0; \\ +\pi_0(U_{\infty}(A)) & n =1. +\end{cases} +\end{equation} + + +\subsection{The pre-determinant} +We will shortly recall the definition of the \emph{pre-determinant} map, which will take a piece-wise smooth path to an element of a real Banach space, when we have a \emph{tracial} map. +To extend its definition to any continuous path, we will need a few results. +The following can be found in the proof of \cite[Lemma 3]{dlHS84a}. + +\begin{lemma}\label{lem:any path homotopic to piecewise smooth} +Any path $\xi \in PU^0(A)$ is homotopic to a piece-wise smooth. +\end{lemma} +In fact, any path $\xi \in PU^0(A)$ is homotopic to a path of the form +\begin{equation} +t \mapsto \xi\left(\frac{j-1}{k}\right)e^{2\pi i(kt -j + 1)a_j}, t \in \left[\frac{j-1}{k},\frac{j}{k}\right], j=1,\dots,k, +\end{equation} +where $k$ is large enough such that +\begin{equation} +\left\| \xi\left(\frac{j-1}{k}\right)^{-1}\xi\left(\frac{j}{k}\right) - 1\right\| < 1 \text{ for all } j=1,\dots,k +\end{equation} +and +\begin{equation} +a_j = \frac{1}{2\pi i}\log \left(\xi\left(\frac{j-1}{k}\right)^{-1}\xi\left(\frac{j}{k}\right)\right) \text{ for } j=1,\dots,k. +\end{equation} +We make the historical note that some of the ideas of working with paths in this way can be found in \cite{ArakiSmithSmith71}. + + +If $\tau: A_{sa} \to E$ is a bounded tracial map to a real Banach $E$ (tracial meaning that $\tau(aa^*) = \tau(a^*a)$ for all $a \in A$), we can associate to a piece-wise smooth path $\xi \in PU^0(A)$ the element +\begin{equation} +\tD_\tau(\xi) := \int_0^1\tau\left(\frac{1}{2\pi i}\xi'(t)\xi(t)^{-1}\right)dt \in E, +\end{equation} +which is just the Riemann path-integral in $E$. +We note that this is well-defined since it is easily seen that, when $\xi'(t)$ is defined, $\xi'(t)\xi(t)^{-1}$ is skew-adjoint, so that $\frac{1}{2\pi i}\xi'(t)\xi(t)^{-1}$ is self-adjoint. +This map also extends to $U_n^0(A)$, hence $U_{\infty}^0(A)$, in the canonical way via the unnormalized trace: for $\xi \in PU_n^0(A)$ piece-wise smooth, write +\begin{equation} +\tD_\tau(\xi) := \int_0^1 \tr_n \otimes \tau \left(\frac{1}{2\pi i}\xi'(t)\xi(t)^{-1}\right)dt, +\end{equation} +where $\tr_n \otimes \tau: M_n(A)_{sa} \to E$ is the (unnormalized) extension of the trace, given by $\tr_n\otimes \tau((a_{ij})) = \sum_i \tau(a_{ii})$, whenever $(a_{ij}) \in M_n(A)_{sa}$. +In this case, we have +\begin{equation} +\tD_\tau(\xi \oplus 1_{m-n}) = \tD_\tau(\xi). +\end{equation} +whenever $\xi \in PU_n^0(A)$ and $m > n$. + +We state the unitary variant of \cite[Lemme 1]{dlHS84a}, suited to our particular needs. + +\begin{lemma}\label{lem:pre-det facts} +Let $\tau: A_{sa} \to E$ be a bounded trace on a unital C*-algebra $A$. The map $\tD$, which takes a piece-wise smooth path $\xi \in PU^0(A)$ (or $\xi \in PU_{\infty}^0(A)$) to $\tD_\tau(\xi)$, has the following properties. +\begin{enumerate} +\item It takes point-wise products to sums: if $\xi_1,\xi_2$ are two piece-wise smooth paths, then +\begin{equation} +\tD_\tau(\xi_1\xi_2) = \tD_\tau(\xi_1) + \tD_\tau(\xi_2), +\end{equation} +where $\xi_1\xi_2$ is the piece-wise smooth path $[\xi_1\xi_2](t) = \xi_1(t)\xi_2(t)$. +\item If $\|\xi(t) - 1\| < 1$ for all $t \in [0,1]$, then +\begin{equation} +\tD_\tau(\xi) = \frac{1}{2\pi i}\tau\left(\log \xi(1)\right). +\end{equation} +\item $\tD_\tau(\xi)$ depends only on the continuous homotopy class of $\xi$ with fixed endpoints. +\item If $p \in A$ is a projection, then the path $\xi_p \in PU^0(A)$ given by +\begin{equation} +\xi_p(t) = pe^{2\pi i t} + (1-p) +\end{equation} +satisfies $\tD_\tau(\xi_p) = \tau(p)$. Note that $\xi_p(t)$ can also be written as $\xi_p(t) = e^{2\pi i tp}$. +\end{enumerate} +\end{lemma} + +By combining Lemma \ref{lem:any path homotopic to piecewise smooth} and Lemma \ref{lem:pre-det facts}(3), it makes sense to extend $\tD_\tau$ to a map +\begin{equation} +\tD_\tau: PU^0(A) \to E +\end{equation} +by $\tD_\tau(\xi) = \tD_\tau(\xi_0)$ where $\xi_0 \in PU^0(A)$ is any piece-wise smooth path homotopic to $\xi$, and this map is clearly a group homomorphism. +Lemma \ref{lem:pre-det facts}(4) in fact allows us to determine the pairing map between the $K_0$-group and traces. + +We will be interested in $\II_1$ von Neumann algebras $\cM$, which have a normal, faithful, positive, center-valued trace \cite[Theorem V.2.6]{TakesakiI}, which we will write as $\Tr_\cM: \cM \to Z(\cM)$. Moreover, as it is self-adjoint in the sense that $\Tr_\cM(\cM_{sa}) \subseteq Z(\cM)_{sa}$, we will abuse notation and write $\Tr_\cM$ for the restriction to $\cM_{sa}$. We will just write $\tD$ for $\tD_{\Tr_\cM}$. +We would like a generalization of the fact that the pairing between $K$-theory and traces of a $\II_1$ factor $\cM$ gives an isomorphism $K_0(\cM) \simeq \bR$. +We note that, for example in \cite{Handelman78}, it was shown that $\pi_1(U(\cM)) \simeq Z(\cM)_{sa}$ in the $\II_1$ von Neumann algebra setting (actually this was shown to hold more generally). We will want to work with a specific isomorphism. + +\begin{lemma}\label{lem:fundamental group is center} +Let $\cM$ be a type $\II_1$ von Neumann algebra. Then +\begin{equation} +\tD|_{\pi_1(U(\cM))}: \pi_1(U(\cM)) \to Z(\cM)_{sa} +\end{equation} +is a group isomorphism. (We are abusing notation by writing $\tD|_{\pi_1(U(\cM))}$, but as $\tD$ is homotopy invariant, it makes sense to apply it to homotopy classes.) +\begin{proof} +As $\cM$ is a $\II_1$ von Neumann algebra, it follows from Corollary 4.5 and Theorem 4.13 of \cite{Rieffel87} that the canonical map +\begin{equation}\label{eq:canonical map is iso} +\pi_1(U(\cM)) \to \pi_1(U_{\infty}(\cM)) = K_0(\cM) +\end{equation} +is an isomorphism. + +Now given some positive contraction $a \in Z(\cM)$, there is a projection $p \in \cM$ such that $\Tr_\cM(p) = a$ by \cite[Theorem 8.4.4(ii)]{KadisonRingroseII}. +As such, this implies that given $n \in \bN$ and any positive element $a \in Z(\cM)$ with $\|a\| \leq n$, we can find a projection in $p \in M_n(\cM)$ such that $\tr_n \otimes \Tr_\cM(p) = a$. +As every self-adjoint element in $Z(\cM)$ is the difference of positive elements in $Z(\cM)$ and Murray-von Neumann equivalent projections $p,q \in M_n(\cM)$ will satisfy $\Tr_\cM(p) = \Tr_\cM(q)$, we get a well-defined group homomorphism +\begin{equation} +K_0(\Tr_\cM): K_0(\cM) \to Z(\cM)_{sa} +\end{equation} +which is surjective. +To show that this map is injective, we simply note that that $p \MvN q$ if and only if $\Tr_\cM(p) = \Tr_\cM(q)$ by \cite[Theorem 8.4.3(iii)]{KadisonRingroseII}. + +Now Lemma \ref{lem:pre-det facts}(4), together with (\ref{eq:canonical map is iso}), allows us to realize this map as precisely the pre-determinant applied to a loop. +\end{proof} +\end{lemma} + +\subsection{Covering spaces} +Finally, let us say something about covering groups. +We use results from \cite[Chapter 10]{RotmanATBook}; one can also see \cite[Chapter 1.3]{HatcherAT}. +Covering spaces are common objects in algebraic topology, and are useful for computing homotopy groups of relatively nice spaces. +For example, one can use the universal covering space of the unit circle to show that its fundamental group can be identified with the integers. + + +A \emph{covering space} of a topological space $X$ consists of a topological space $\tilde{X}$ and a continuous surjective map $p: \tilde{X} \to X$ such that for each $x \in X$, there is an open neighbourhood $U$ of $x$ in $X$ such that $p^{-1}(U)$ is a union of disjoint open sets, each of which is mapped homeomorphically onto $U$ by $p$. +A covering space $\tilde{X}$ of $X$ is a \emph{universal covering space} if $\tilde{X}$ is simply connected; that is, it is path-connected with trivial fundamental group. + + \begin{example} + The real numbers $\bR$ is the universal covering space of the unit circle $\bT$ with covering map $p(t) = e^{2\pi i t}$. In fact, this generalizes to the following: $SU(n) \times \bR$ is a cover of $U(n)$. + Indeed, we have covers + \begin{equation} + SU(n) \times \bR \to SU(n) \times \bT \to U(n) + \end{equation} + where the first map is $\id \times p$, where $p$ is the covering map $\bR \to \bT$ coming from the universal cover of $\bT$, and $SU(n) \times \bT \to U(n)$ is the map $(U,\lambda) \mapsto \lambda U$. + As $SU(n) \times \bR$ is simply connected, it follows that this is the universal cover of $U(n)$. + \end{example} + + For a topological group $G$, a \emph{covering group} $\widetilde{G}$ of $G$ is a covering space of $G$ such that $\widetilde{G}$ is a topological group and the covering map $\rho: \widetilde{G} \to G$ is a continuous group homomorphism. + A covering group is the \emph{universal covering group} if it is universal as a covering space. In fact, covering spaces of topological groups can be equipped with a \emph{some} multiplication which make them into a covering group \cite[Theorem 10.42]{RotmanATBook}. + + A topological space $X$ is said to be \emph{locally path-connected} if every neighbourhood of every point contains a path-connected neighbourhood and \emph{semilocally simply connected} if each point has a neighbourhood $N$ with the property that every loop in it can be contracted to a single point within $X$ (i.e., every loop in $N$ is null-homotopic in $X$). + + \begin{theorem} + A topological group $G$ which is path-connected, locally path-connected, and semilocally simply connected has a universal covering $\widetilde{G}$, unique up to homeomorphic isomorphism. Moreover, $\tilde{G}$ can be realized as + \begin{equation} + \widetilde{G} = PG/\sim_{nh} + \end{equation} + where $\sim_{nh}$ is the equivalence relation of null-homotopy. + \end{theorem} + As mentioned in the introdction, the universal covering group of $U(n)$ is a semidriect product $SU(n) \rtimes \bR$. + + + + When a topological group $G$ has a universal covering group, we obtain a short exact sequence + \begin{equation}\label{eq:covering group ses} + \begin{tikzcd} +1 \arrow[r] & \pi_1(G) \arrow[r] & \widetilde{G} \arrow[r] & G \arrow[r] & 1 +\end{tikzcd} +\end{equation} +where the first map is just the canonical embedding of homotopy classes into $\tilde{G}$, and the second map is evaluation at 1. + + + +\section{The universal covering group of $U(\cM)$}\label{sec:universal cover of U(M)} + +We have spoken about the universal covering group of $U(n)$ in the introduction, so let us move on to the unitary groups $U(\cM)$ where $\cM$ is not of type $I_n$. +In the case where $\cM$ is a properly infinite von Neumann algebra, the unitary group is path-connected and locally path-connected by the arguments below. For the last condition, one can appeal to \cite[Theorem 3.5]{Handelman78} to see that $\pi_1(U(\cM)) = 0$, i.e., that $U(\cM)$ is simply connected and therefore semilocally simply connected (as it is connected). +As such, the universal covering group of $U(\cM)$ exists, but it is trivially just $U(\cM)$ due to the fact that $\pi_1(U(\cM)) = 0$. + +Let us first prove that the universal covering group of $U(\cM)$ exists whenever $\cM$ is a type $\II_1$ von Neumann algebra. + +\begin{lemma} +Let $\cM$ be a $\II_1$ von Neumann algebra. Then $U(\cM)$ is path-connected, locally path-connected, and semi-locally simply connected. In particular, the universal covering group of $U(\cM)$ exists and can be identified with +\begin{equation} +PU(\cM)/\sim_{nh}. +\end{equation} +\begin{proof} +Clearly $U(\cM)$ is path-connected as every unitary in a von Neumann algebra can be written as an exponential $u = e^{2\pi i a}$ for some self-adjoint $a \in A$ (by using functional calculus), and so the path $t \mapsto e^{2\pi i ta}$ yields a path from 1 to $u$. +The second condition amounts to showing that every open neighbourhood of an element contains a path-connected open neighbourhood. +This is trivial because if $\|u - v\| < 2$, then $u,v$ are path-connected (see, for example, \cite[Lemma 2.1.3(iii)]{RordamKBook}). +Thus if $N$ is a neighbourhood of $u \in U(\cM)$, let $0 < \ee < 1$ be such that $B_\ee(u) \subseteq N$. +Then any two points in $B_\ee(u)$ will be within distance 2 of each other. + +The third condition says that every element $u \in U(\cM)$ has a neighbourhood $N$ such that every loop in $N$ is null-homotopic to a point in $U(\cM)$. To see this, let $u \in U(\cM)$ and consider $N = B_\frac{1}{2}(u)$, the open ball of radius $\frac{1}{2}$ around $u$. Take some $u_0 \in N$ and suppose that $\xi: [0,1] \to N$ is a loop with initial and terminal point $u_0$. +We note that +\begin{equation} +\|\xi(t) - u_0\| \leq \|\xi(t) - u\| + \|u - u_0\| < \frac{1}{2} + \frac{1}{2} = 1. +\end{equation} +Define $\xi_0(t) = u_0^*\xi(t)$, which is a loop with initial and terminal point $1_\cM$, and note that +\begin{equation} +\begin{split} +\|\xi_0(t) - 1\| &= \|u_0^*\xi(t) - 1\| \\ +&= \|\xi(t) - u_0\| < 1. +\end{split} +\end{equation} +We can now apply \ref{lem:pre-det facts}(2) to get that +\begin{equation} +\tD_\tau(\xi_0) = \frac{1}{2\pi i}\Tr_\cM\left(\log\xi_0(1)\right) = \frac{1}{2\pi i}\Tr_\cM(\log 1) = 0. +\end{equation} +In particular, as $\tD|_{\pi_1(U(\cM))}: \pi_1(U(\cM)) \to Z(\cM)_{sa}$ is an isomorphism, we have that $\xi_0$ is null-homotopic. +But then $\xi = u_0\xi_0$ is homotopic to the constant path $t \mapsto u_0$. +\end{proof} +\end{lemma} + +\begin{remark} +Suppose that $A$ is a unital C*-algebra and $\tD: PU^0(A) \to \Aff T(A)$ is the pre-determinant associated to the tracial map $\Tr_A: A_{sa} \to A_{sa}/A_0$ (which is just the quotient map) where $A_0$ is the subspace of all self-adjoint elements which vanish on every tracial state. $\Tr_A$ is often referred to as the universal trace. +If $\tD$ ``determines homotopy'' in the sense that whenever $\xi$ is a loop, we have that $\tD(\xi) = 0$ if and only if $\xi$ is null-homotopic, then the same argument above will yield that the universal covering group of $U^0(A)$ will exist. + +Thus whenever $A$ is $K_1$-injective, has trivial $K_1$ group, and satisfies the above property of the pre-determinant realizing homotopy, then $U(A)$ will have a universal cover. +For example, the unitary group of the Jiang-Su algebra $\cZ$ \cite{JiangSu99} or the unitary group of a UHF algebra will have a universal cover. +\end{remark} + +Recall that a short exact sequence +\begin{equation} +\begin{tikzcd} +1 \arrow[r] & K \arrow[r, "\alpha"] & S \arrow[r, "\beta"] & U \arrow[r] & 1 +\end{tikzcd} +\end{equation} +of groups splits if there is a group homomorphism $\sigma: U \to S$ such that $\beta \circ \sigma = \id_U$, and this happens if and only if $S = \alpha(K) \rtimes U'$, where $U' \leq S$ is some copy of $U$ (namely $\sigma(U)$) with $\beta|_{U'} \to U$ being an isomorphism. Moreover, this is in fact a direct product if and only if this copy $U'$ is normal in $S$. We will need the following well-known fact. + +\begin{lemma}\label{lem:left split ses} +Let +\begin{equation} +\begin{tikzcd} +1 \arrow[r] & K \arrow[r, "\alpha"] & S \arrow[r, "\beta"] \arrow[l, "\gamma", bend left=49] & U \arrow[r] & 1 +\end{tikzcd} +\end{equation} +be a short exact sequence of groups with $\gamma \circ \alpha = \id_K$. Then $S \simeq K \times U$. +\end{lemma} + +\begin{theorem} +Let $\cM$ be a $\II_1$ von Neumann algebra and consider the universal covering group +\begin{equation}\label{eq:universal cover of U} +\begin{tikzcd} +1 \arrow[r] & \pi_1(U(\cM)) \arrow[r] & \widetilde{U(\cM)} \arrow[r] & U(\cM) \arrow[r] & 1. +\end{tikzcd} +\end{equation} +The sequence (\ref{eq:universal cover of U}) splits algebraically, giving +\begin{equation} +\widetilde{U(\cM)} \simeq \pi_1(U(\cM)) \times U(\cM). +\end{equation} +In particular, +\begin{equation}\label{eq:cg is Z times U} +\widetilde{U(\cM)} \simeq Z(\cM)_{sa} \times U(\cM). +\end{equation} +\begin{proof} +Let us denote the map $\pi_1(U(\cM)) \to \widetilde{U(\cM)}$ by $\iota$ and note that $\iota([\xi]) = [\xi]$, i.e., it just takes the homotopy class of $\xi$ to the class of $\xi$ under null-homotopy. +We will write $\ev_1: \widetilde{U(\cM)} \to U(\cM)$ to be evaluation at 1. +By Lemma \ref{lem:fundamental group is center}, we can take the perspective that $K_0(\cM)$ is $\pi_1(U(\cM))$ with +\begin{equation} +\tD|_{\pi_1(U(\cM))}: \pi_1(U(\cM)) \to Z(\cM)_{sa} +\end{equation} +being an isomorphism. + +Define $\gamma: \widetilde{U(\cM)} \to \pi_1(U(\cM))$ via the composition +\begin{equation} +\begin{tikzcd} +\widetilde{U(\cM))} \arrow[r, "\tD_0"] & Z(\cM)_{sa} \arrow[r, "(\tD|_{\pi_1(U(\cM))})^{-1}"] & \pi_1(U(\cM)) +\end{tikzcd} +\end{equation} +where $\tD_0: \widetilde{U(\cM)} \to Z(\cM)_{sa}$ is the map given by $[\xi] \mapsto \tD(\xi)$, which is well-defined since $\tD$ preserves homotopy classes of paths with the same endpoints. +That is, given some class $[\xi]$ of a path $[0,1] \to U(\cM)$, up to null-homotopy, we evaluate the pre-determinant at this (homotopy class of a) path which gives us a self-adjoint element of the center. We then find a loop, up to homotopy, which gives that same self-adjoint element when we evaluate the pre-determinant. But then we realize that +\begin{equation} +\gamma \circ \iota([\xi]) = \gamma([\xi]) = [\xi] \text{ for } [\xi] \in \pi_1(U(\cM)), +\end{equation} +i.e., $\gamma \circ \iota = \id_{\pi_1(U(\cM))}$. +So we have the left split short exact sequence +\begin{equation} +\begin{tikzcd} +1 \arrow[r] & \pi_1(U(\cM)) \arrow[r, "\iota"] & \widetilde{U(\cM)} \arrow[r, "\ev_1"] \arrow[l, "\gamma", bend left=49] & U(\cM) \arrow[r] & 1, +\end{tikzcd} +\end{equation} +which gives that $\widetilde{U(\cM)} \simeq \pi_1(U(\cM)) \times U(\cM)$ by Lemma \ref{lem:left split ses}. Now (\ref{eq:cg is Z times U}) follows from our identification of $\pi_1(U(\cM))$ with $Z(\cM)_{sa}$. +\end{proof} +\end{theorem} + +\begin{remark} +Even though we have a direct product as groups, it is clear that this is not a direct product as topological spaces. Indeed, a universal cover must be simply connected. This clearly fails to be true if $Z(\cM)_{sa}$ and $U(\cM)$ were equipped with their (relative) norm topologies. +Moreover, $\pi_1(U(\cM)) \subseteq \widetilde{U(\cM)}$ is considered as a discrete set. +\end{remark} + +Recall that a group $G$ is perfect if $G = DG$, i.e., every element is a finite product of commutators. +With the above, we conclude that the universal covering group of $U(\cM)$ is not perfect. + + +\begin{cor} +The universal covering group of $U(\cM)$ is not perfect whenever $\cM$ is a $\II_1$ von Neumann algebra. +\begin{proof} +If $G = A \times H$ where $H$ is some group and $A$ is abelian, we have that $DG = DA \times DH = \{0\} \times DH$. So if $|A| > 1$, then $G$ can never be perfect. +\end{proof} +\end{cor} + + + + \bibliographystyle{amsalpha} + \bibliography{biblio} + +\end{document} diff --git a/Universal covering groups of unitary groups of von Neumann algebras/Universal_cover_of_U_M.toc b/Universal covering groups of unitary groups of von Neumann algebras/Universal_cover_of_U_M.toc new file mode 100644 index 0000000..b51a09c --- /dev/null +++ b/Universal covering groups of unitary groups of von Neumann algebras/Universal_cover_of_U_M.toc @@ -0,0 +1,11 @@ +\babel@toc {nil}{}\relax +\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{section.1}% +\setcounter {tocdepth}{0} +\contentsline {section}{\tocsection {}{}{Acknowledgements}}{2}{section*.2}% +\setcounter {tocdepth}{1} +\contentsline {section}{\tocsection {}{2}{Preliminaries}}{3}{section.2}% +\contentsline {subsection}{\tocsubsection {}{2.1}{Notation}}{3}{subsection.2.1}% +\contentsline {subsection}{\tocsubsection {}{2.2}{The pre-determinant}}{4}{subsection.2.2}% +\contentsline {subsection}{\tocsubsection {}{2.3}{Covering spaces}}{6}{subsection.2.3}% +\contentsline {section}{\tocsection {}{3}{The universal covering group of $U(\mathcal {M})$}}{7}{section.3}% +\contentsline {section}{\tocsection {}{}{References}}{9}{section*.3}% diff --git a/Universal covering groups of unitary groups of von Neumann algebras/biblio.bib b/Universal covering groups of unitary groups of von Neumann algebras/biblio.bib new file mode 100644 index 0000000..4d7df5a --- /dev/null +++ b/Universal covering groups of unitary groups of von Neumann algebras/biblio.bib @@ -0,0 +1,4920 @@ +### Other +@book {IssaiSchur, + TITLE = {Studies in memory of {I}ssai {S}chur}, + SERIES = {Progress in Mathematics}, + VOLUME = {210}, + EDITOR = {Joseph, Anthony and Melnikov, Anna and Rentschler, Rudolf}, + NOTE = {Papers from the Paris Midterm Workshop of the European + Community Training and Mobility of Researchers (TMR) Network + held in Chevaleret, May 21--25, 2000 and the Schur Memoriam + Workshop held in Rehovot, December 27--31, 2000}, + PUBLISHER = {Birkh\"{a}user Boston, Inc., Boston, MA}, + YEAR = {2003}, + PAGES = {clxxxvii+365}, + ISBN = {0-8176-4208-0}, + MRCLASS = {20-06 (00B30 20-03)}, + MRNUMBER = {1985184}, + DOI = {10.1007/978-1-4612-0045-1}, + URL = {https://doi.org/10.1007/978-1-4612-0045-1}, +} + + +#### AAAAAA + +# Aguilar-Socolovsky +@article {AguilarSocolovsky00, + AUTHOR = {Aguilar, Marcelo A. and Socolovsky, Miguel}, + TITLE = {Universal covering group of {${\rm U}(n)$} and projective + representations}, + JOURNAL = {Internat. J. Theoret. Phys.}, + FJOURNAL = {International Journal of Theoretical Physics}, + VOLUME = {39}, + YEAR = {2000}, + NUMBER = {4}, + PAGES = {997--1013}, + ISSN = {0020-7748,1572-9575}, + MRCLASS = {81R05 (22E70 81-01)}, + MRNUMBER = {1779169}, +MRREVIEWER = {S.\ Timothy\ Swift}, + DOI = {10.1023/A:1003694206391}, + URL = {https://doi.org/10.1023/A:1003694206391}, +} + + +# Albert-Muckenhoupt +@article{AlbertMuckenhoupt57, + title={On matrices of trace zero}, + author={Albert, Abraham A. and Muckenhoupt, Benjamin}, + journal={Michigan Mathematical Journal}, + volume={4}, + number={1}, + pages={1--3}, + year={1957}, + publisher={University of Michigan, Department of Mathematics} +} + +# Akemann-Anderson-Pedersen +@article {AkemannAndersonPedersen86, + AUTHOR = {Akemann, Charles A. and Anderson, Joel and Pedersen, Gert K.}, + TITLE = {Excising states of {C}*-algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {38}, + YEAR = {1986}, + NUMBER = {5}, + PAGES = {1239--1260}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (46L30)}, + MRNUMBER = {869724}, +MRREVIEWER = {J. W. Bunce}, + DOI = {10.4153/CJM-1986-063-7}, + URL = {https://doi.org/10.4153/CJM-1986-063-7}, +} + +# Al-Rawashdeh-Booth-Giordano +@article {AlBoothGiordano12, + AUTHOR = {Al-Rawashdeh, Ahmed and Booth, Andrew and Giordano, Thierry}, + TITLE = {Unitary groups as a complete invariant}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {262}, + YEAR = {2012}, + NUMBER = {11}, + PAGES = {4711--4730}, + ISSN = {0022-1236}, + MRCLASS = {46L35}, + MRNUMBER = {2913684}, + DOI = {10.1016/j.jfa.2012.03.016}, + URL = {https://doi.org/10.1016/j.jfa.2012.03.016}, +} + +# Alekseev-Schmidt-Thom +@article{AlekseevSchmidtThom23, + title={Amenability for unitary groups of {C}*-algebras}, + author={Alekseev, Vadim and Schmidt, Max and Thom, Andreas}, + journal={arXiv:2305.13181}, + year={2023} +} + +# Amini-Golestani-Jamali-Phillips +@article{AGJP22, + title={Group actions on simple tracially $\mathcal{Z}$-absorbing {C}*-algebras}, + author={Amini, Massoud and Golestani, Nasser and Jamali, Saeid and Phillips, N. Christopher}, + journal={arXiv:2204.03615}, + year={2022}, +} + +# Amrutam-Kalantar +@article {AmrutamKalantar20, + AUTHOR = {Amrutam, Tattwamasi and Kalantar, Mehrdad}, + TITLE = {On simplicity of intermediate {C}*-algebras}, + JOURNAL = {Ergodic Theory Dynam. Systems}, + FJOURNAL = {Ergodic Theory and Dynamical Systems}, + VOLUME = {40}, + YEAR = {2020}, + NUMBER = {12}, + PAGES = {3181--3187}, + ISSN = {0143-3857}, + MRCLASS = {46L55 (37A55 46L10 46L35 46L89)}, + MRNUMBER = {4170599}, +MRREVIEWER = {Bruno Brogni Uggioni}, + DOI = {10.1017/etds.2019.34}, + URL = {https://doi.org/10.1017/etds.2019.34}, +} + +# Ando-Haagerup +@article {AndoHaagerup14, + AUTHOR = {Ando, Hiroshi and Haagerup, Uffe}, + TITLE = {Ultraproducts of von {N}eumann algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {266}, + YEAR = {2014}, + NUMBER = {12}, + PAGES = {6842--6913}, + ISSN = {0022-1236}, + MRCLASS = {46M07 (46L10)}, + MRNUMBER = {3198856}, +MRREVIEWER = {E. St{\o}rmer}, + DOI = {10.1016/j.jfa.2014.03.013}, + URL = {https://doi.org/10.1016/j.jfa.2014.03.013}, +} + +# Antoine-Perera-Santiago +@article {AntoinePereraSantiago11, + AUTHOR = {Antoine, Ramon and Perera, Francesc and Santiago, Luis}, + TITLE = {Pullbacks, {$C(X)$}-algebras, and their {C}untz semigroup}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {260}, + YEAR = {2011}, + NUMBER = {10}, + PAGES = {2844--2880}, + ISSN = {0022-1236,1096-0783}, + MRCLASS = {46L35 (46L05 46L80 46M15)}, + MRNUMBER = {2774057}, +MRREVIEWER = {Valentin\ Deaconu}, + DOI = {10.1016/j.jfa.2011.02.016}, + URL = {https://doi.org/10.1016/j.jfa.2011.02.016}, +} + +# Ara-Mathieu +@book {AraMathieubook, + AUTHOR = {Ara, Pere and Mathieu, Martin}, + TITLE = {Local multipliers of {C}*-algebras}, + SERIES = {Springer Monographs in Mathematics}, + PUBLISHER = {Springer-Verlag London, Ltd., London}, + YEAR = {2003}, + PAGES = {xii+319}, + ISBN = {1-85233-237-9}, + MRCLASS = {46L05 (16S90 46Kxx 46L40 47B47 47L25)}, + MRNUMBER = {1940428}, +MRREVIEWER = {Michael Frank}, + DOI = {10.1007/978-1-4471-0045-4}, + URL = {https://doi.org/10.1007/978-1-4471-0045-4}, +} + +# Araki-Smith-Smith +@article {ArakiSmithSmith71, + AUTHOR = {Araki, Huzihiro and Smith, Mi-Soo B. and Smith, Larry}, + TITLE = {On the homotopical significance of the type of von {N}eumann + algebra factors}, + JOURNAL = {Comm. Math. Phys.}, + FJOURNAL = {Communications in Mathematical Physics}, + VOLUME = {22}, + YEAR = {1971}, + PAGES = {71--88}, + ISSN = {0010-3616,1432-0916}, + MRCLASS = {46.65 (57.00)}, + MRNUMBER = {288587}, +MRREVIEWER = {W.\ Wils}, + URL = {http://projecteuclid.org/euclid.cmp/1103857414}, +} + +# Argerami-Farenick +@article {ArgeramiFarenick08, + AUTHOR = {Argerami, Mart\'{\i}n and Farenick, Douglas R.}, + TITLE = {Local multiplier algebras, injective envelopes, and type {I} + {W}*-algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {59}, + YEAR = {2008}, + NUMBER = {2}, + PAGES = {237--245}, + ISSN = {0379-4024,1841-7744}, + MRCLASS = {46L05 (46L07)}, + MRNUMBER = {2411044}, +MRREVIEWER = {Pere\ Ara}, +} + +# Avitzer +@article {Avitzer82, + AUTHOR = {Avitzour, Daniel}, + TITLE = {Free products of {C}*-algebras}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {271}, + YEAR = {1982}, + NUMBER = {2}, + PAGES = {423--435}, + ISSN = {0002-9947,1088-6850}, + MRCLASS = {46L05}, + MRNUMBER = {654842}, +MRREVIEWER = {William\ Paschke}, + DOI = {10.2307/1998890}, + URL = {https://doi.org/10.2307/1998890}, +} + +#### BBBBBBBBBBBBBBBBBBBBBBBB +@book {BerberianBook, + AUTHOR = {Berberian, Sterling K.}, + TITLE = {Baer *-rings}, + SERIES = {Die Grundlehren der mathematischen Wissenschaften}, + VOLUME = {Band 195}, + PUBLISHER = {Springer-Verlag, New York-Berlin}, + YEAR = {1972}, + PAGES = {xiii+296}, + MRCLASS = {16A28 (46L10)}, + MRNUMBER = {429975}, +MRREVIEWER = {S.\ S.\ Holland, Jr.}, +} + +#Bisch +@article {Bisch90, + AUTHOR = {Bisch, Dietmar H.}, + TITLE = {On the existence of central sequences in subfactors}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {321}, + YEAR = {1990}, + NUMBER = {1}, + PAGES = {117--128}, + ISSN = {0002-9947}, + MRCLASS = {46L35}, + MRNUMBER = {1005075}, +MRREVIEWER = {V. S. Sunder}, + DOI = {10.2307/2001593}, + URL = {https://doi.org/10.2307/2001593}, +} + +@article {Bisch94, + AUTHOR = {Bisch, Dietmar H.}, + TITLE = {Central sequences in subfactors. {II}}, + JOURNAL = {Proc. Amer. Math. Soc.}, + FJOURNAL = {Proceedings of the American Mathematical Society}, + VOLUME = {121}, + YEAR = {1994}, + NUMBER = {3}, + PAGES = {725--731}, + ISSN = {0002-9939}, + MRCLASS = {46L37}, + MRNUMBER = {1209417}, +MRREVIEWER = {V. S. Sunder}, + DOI = {10.2307/2160268}, + URL = {https://doi.org/10.2307/2160268}, +} + +# Berrick +@article {Berrick11, + AUTHOR = {Berrick, A. Jon}, + TITLE = {The acyclic group dichotomy}, + JOURNAL = {J. Algebra}, + FJOURNAL = {Journal of Algebra}, + VOLUME = {326}, + YEAR = {2011}, + PAGES = {47--58}, + ISSN = {0021-8693,1090-266X}, + MRCLASS = {20F38 (19E20 20J05 57M07)}, + MRNUMBER = {2746051}, +MRREVIEWER = {Aditi\ Kar}, + DOI = {10.1016/j.jalgebra.2010.05.009}, + URL = {https://doi.org/10.1016/j.jalgebra.2010.05.009}, +} + + +# Blackadar +@article {Blackadar90, + AUTHOR = {Blackadar, Bruce}, + TITLE = {Symmetries of the {CAR} algebra}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {131}, + YEAR = {1990}, + NUMBER = {3}, + PAGES = {589--623}, + ISSN = {0003-486X}, + MRCLASS = {46L80 (19K14)}, + MRNUMBER = {1053492}, +MRREVIEWER = {Bola O. Balogun}, + DOI = {10.2307/1971472}, + URL = {https://doi.org/10.2307/1971472}, +} + +@book {BlackadarBook, + AUTHOR = {Blackadar, Bruce}, + TITLE = {Operator algebras}, + SERIES = {Encyclopaedia of Mathematical Sciences}, + VOLUME = {122}, + NOTE = {Theory of {C}*-algebras and von {N}eumann algebras, + Operator Algebras and Non-commutative Geometry, {III}}, + PUBLISHER = {Springer-Verlag, Berlin}, + YEAR = {2006}, + PAGES = {xx+517}, + ISBN = {978-3-540-28486-4; 3-540-28486-9}, + MRCLASS = {46L05 (46L10 46L80)}, + MRNUMBER = {2188261}, +MRREVIEWER = {Paul Jolissaint}, + DOI = {10.1007/3-540-28517-2}, + URL = {https://doi.org/10.1007/3-540-28517-2}, +} + +# Blacakdar-Kumjian-Rordam +@article {BKR92, + AUTHOR = {Blackadar, Bruce and Kumjian, Alexander and R{\o}rdam, Mikael}, + TITLE = {Approximately central matrix units and the structure of + noncommutative tori}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {6}, + YEAR = {1992}, + NUMBER = {3}, + PAGES = {267--284}, + ISSN = {0920-3036}, + MRCLASS = {46L87 (46L05)}, + MRNUMBER = {1189278}, +MRREVIEWER = {Chi Wai Leung}, + DOI = {10.1007/BF00961466}, + URL = {https://doi.org/10.1007/BF00961466}, +} + +# Booth +@mastersthesis{Booth98, + title={The unitary group as a complete invariant for simple unital {AF} algebras}, + author={Booth, Andrew}, + year={1998}, + school={University of Ottawa (Canada)}, +} + +# Bosa-Gabe-Sims-White +@article {BGSW22, + AUTHOR = {Bosa, Joan and Gabe, James and Sims, Aidan and White, Stuart}, + TITLE = {The nuclear dimension of $\mathcal{O}_\infty$-stable {C}*-algebras}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {401}, + YEAR = {2022}, + PAGES = {Paper No. 108250, 51}, + ISSN = {0001-8708}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {4392219}, + DOI = {10.1016/j.aim.2022.108250}, + URL = {https://doi.org/10.1016/j.aim.2022.108250}, +} + +# Brattelo-Stormer-Kishimoto-Rordam +@article {BSKR93, + AUTHOR = {Bratteli, Ola and St{\o}rmer, Erling and Kishimoto, Akitaka and + R{\o}rdam, Mikael}, + TITLE = {The crossed product of a {UHF} algebra by a shift}, + JOURNAL = {Ergodic Theory Dynam. Systems}, + FJOURNAL = {Ergodic Theory and Dynamical Systems}, + VOLUME = {13}, + YEAR = {1993}, + NUMBER = {4}, + PAGES = {615--626}, + ISSN = {0143-3857}, + MRCLASS = {46L55 (46L35)}, + MRNUMBER = {1257025}, +MRREVIEWER = {Robert J. Archbold}, + DOI = {10.1017/S0143385700007574}, + URL = {https://doi.org/10.1017/S0143385700007574}, +} + +#Bresar +@article {Bresar93, + AUTHOR = {Bre\v{s}ar, Matej}, + TITLE = {Commuting traces of biadditive mappings, + commutativity-preserving mappings and {L}ie mappings}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {335}, + YEAR = {1993}, + NUMBER = {2}, + PAGES = {525--546}, + ISSN = {0002-9947}, + MRCLASS = {16W25 (16N60 16W10)}, + MRNUMBER = {1069746}, +MRREVIEWER = {Howard E. Bell}, + DOI = {10.2307/2154392}, + URL = {https://doi.org/10.2307/2154392}, +} + +# Breuer +@article {Breuer70, + AUTHOR = {Breuer, M.}, + TITLE = {On the homotopy type of the group of regular elements of + semifinite von {N}eumann algebras}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {185}, + YEAR = {1970}, + PAGES = {61--74}, + ISSN = {0025-5831,1432-1807}, + MRCLASS = {46.65 (57.55)}, + MRNUMBER = {264408}, +MRREVIEWER = {Kazimierz\ G\polhk eba}, + DOI = {10.1007/BF01350761}, + URL = {https://doi.org/10.1007/BF01350761}, +} + +# Breuillard-Kalantar-Kennedy-Ozawa +@article {BKKO17, + AUTHOR = {Breuillard, Emmanuel and Kalantar, Mehrdad and Kennedy, + Matthew and Ozawa, Narutaka}, + TITLE = {{$C^*$}-simplicity and the unique trace property for discrete + groups}, + JOURNAL = {Publ. Math. Inst. Hautes \'Etudes Sci.}, + FJOURNAL = {Publications Math\'ematiques. Institut de Hautes \'Etudes + Scientifiques}, + VOLUME = {126}, + YEAR = {2017}, + PAGES = {35--71}, + ISSN = {0073-8301,1618-1913}, + MRCLASS = {46L10 (20C07 37A55 46L89)}, + MRNUMBER = {3735864}, +MRREVIEWER = {Anton\ Deitmar}, + DOI = {10.1007/s10240-017-0091-2}, + URL = {https://doi.org/10.1007/s10240-017-0091-2}, +} + +# Brodzku-Cave-Li +@article {BrodzkiCaveLi17, + AUTHOR = {Brodzki, Jacek and Cave, Chris and Li, Kang}, + TITLE = {Exactness of locally compact groups}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {312}, + YEAR = {2017}, + PAGES = {209--233}, + ISSN = {0001-8708,1090-2082}, + MRCLASS = {46L10 (19K56 22D25)}, + MRNUMBER = {3635811}, +MRREVIEWER = {Judith\ A.\ Packer}, + DOI = {10.1016/j.aim.2017.03.020}, + URL = {https://doi.org/10.1016/j.aim.2017.03.020}, +} + +# Broise +@article {Broise67, + AUTHOR = {Broise, Michel}, + TITLE = {Commutateurs dans le groupe unitaire d'un facteur}, + JOURNAL = {J. Math. Pures Appl. (9)}, + FJOURNAL = {Journal de Math\'ematiques Pures et Appliqu\'ees. Neuvi\`eme + S\'erie}, + VOLUME = {46}, + YEAR = {1967}, + PAGES = {299--312}, + ISSN = {0021-7824,1776-3371}, + MRCLASS = {46.65 (47.00)}, + MRNUMBER = {223900}, +MRREVIEWER = {C.\ R.\ Putnam}, +} + +# Brown +@article {Brown81, + AUTHOR = {Brown, Lawrence G.}, + TITLE = {Ext of certain free product {C}*-algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {6}, + YEAR = {1981}, + NUMBER = {1}, + PAGES = {135--141}, + ISSN = {0379-4024}, + MRCLASS = {46M20 (46L05)}, + MRNUMBER = {637007}, +MRREVIEWER = {Claude\ Schochet}, +} + +# Brown-Ozawa +@book {BrownOzawa, + AUTHOR = {Brown, Nathanial P. and Ozawa, Narutaka}, + TITLE = {{C}*-algebras and finite-dimensional approximations}, + SERIES = {Graduate Studies in Mathematics}, + VOLUME = {88}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {2008}, + PAGES = {xvi+509}, + ISBN = {978-0-8218-4381-9; 0-8218-4381-8}, + MRCLASS = {46L05 (43A07 46-02 46L10)}, + MRNUMBER = {2391387}, +MRREVIEWER = {Mikael R{\o}rdam}, + DOI = {10.1090/gsm/088}, + URL = {https://doi.org/10.1090/gsm/088}, +} + +@book {BrownBook82, + AUTHOR = {Brown, Kenneth S.}, + TITLE = {Cohomology of groups}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {87}, + PUBLISHER = {Springer-Verlag, New York-Berlin}, + YEAR = {1982}, + PAGES = {x+306}, + ISBN = {0-387-90688-6}, + MRCLASS = {20-02 (18-01 20F32 20J05 55-01)}, + MRNUMBER = {672956}, +MRREVIEWER = {Ross\ Staffeldt}, +} + +#Bruning-Willgerodt +@article {BruningWillgerodt76, + AUTHOR = {Br\"uning, Jochen and Willgerodt, Wolfgang}, + TITLE = {Eine {V}erallgemeinerung eines {S}atzes von {N}. {K}uiper}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {220}, + YEAR = {1976}, + NUMBER = {1}, + PAGES = {47--58}, + ISSN = {0025-5831,1432-1807}, + MRCLASS = {58B05 (46L10 58D05)}, + MRNUMBER = {405483}, +MRREVIEWER = {P.\ A.\ Kuchment}, + DOI = {10.1007/BF01354528}, + URL = {https://doi.org/10.1007/BF01354528}, +} + + +# Bunce +@article {Bunce72, + AUTHOR = {Bunce, John}, + TITLE = {Characterizations of amenable and strongly amenable {C}*-algebras}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {43}, + YEAR = {1972}, + PAGES = {563--572}, + ISSN = {0030-8730}, + MRCLASS = {46L05}, + MRNUMBER = {320764}, +MRREVIEWER = {B. E. Johnson}, + URL = {http://projecteuclid.org/euclid.pjm/1102959351}, +} + +# Burnstein +@article {Burstein10, + AUTHOR = {Burstein, Richard D.}, + TITLE = {Commuting square subfactors and central sequences}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {21}, + YEAR = {2010}, + NUMBER = {1}, + PAGES = {117--131}, + ISSN = {0129-167X}, + MRCLASS = {46L37}, + MRNUMBER = {2642989}, +MRREVIEWER = {Toshihiko Masuda}, + DOI = {10.1142/S0129167X10005945}, + URL = {https://doi.org/10.1142/S0129167X10005945}, +} + +### CCCCCCCCCCCCCCCCCCCCCC + +# Cameron-Smith +@article {CameronSmith19, + AUTHOR = {Cameron, Jan and Smith, Roger R.}, + TITLE = {A {G}alois correspondence for reduced crossed products of + simple {C}*-algebras by discrete groups}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {71}, + YEAR = {2019}, + NUMBER = {5}, + PAGES = {1103--1125}, + ISSN = {0008-414X}, + MRCLASS = {46L55 (46L40)}, + MRNUMBER = {4010423}, +MRREVIEWER = {Robert S. Doran}, + DOI = {10.4153/cjm-2018-014-6}, + URL = {https://doi.org/10.4153/cjm-2018-014-6}, +} + +# Carrion-Castillejos-Evington-Gabe-Schafhauser-Tikuisis-White +@misc{CCEGSTW, + title={Tracially complete {C}*-algebras}, + author={Carri{\'o}n, Jos{\'e} and Castillejos, Jorge and Evington, Samuel and Gabe, James and Schafhauser, Christopher and Tikuisis, Aaron and White, Stuart}, + note={Manuscript in preparation.} +} + +#Carrion-Gabe-Schafhauser-Tikuisis-White +@article{CGSTW23, + title={Classifying *-homomorphisms {I}: unital simple nuclear {C}*-algebras}, + author={Carri{\'o}n, Jos{\'e} R. and Gabe, James and Schafhauser, Christopher and Tikuisis, Aaron, and White, Stuart}, + journal={arXiv:2307.06480}, + year={2023} +} + + +# Castillejos-Evington-Tikuisis-White-Winter +@article {CETWW21, + AUTHOR = {Castillejos, Jorge and Evington, Samuel and Tikuisis, Aaron + and White, Stuart and Winter, Wilhelm}, + TITLE = {Nuclear dimension of simple {C}*-algebras}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {224}, + YEAR = {2021}, + NUMBER = {1}, + PAGES = {245--290}, + ISSN = {0020-9910}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {4228503}, +MRREVIEWER = {Changguo Wei}, + DOI = {10.1007/s00222-020-01013-1}, + URL = {https://doi.org/10.1007/s00222-020-01013-1}, +} + +# Castillejos-Evington-Tikuisis-White +@article{CETW21, + author = {Castillejos, Jorge and Evington, Samuel and Tikuisis, Aaron and White, Stuart}, + title = "{Uniform Property $\Gamma$}", + journal = {International Mathematics Research Notices}, + volume = {2022}, + number = {13}, + pages = {9864-9908}, + year = {2021}, + month = {02}, + abstract = "{We further examine the concept of uniform property \\$\\Gamma \\$ for \\$C^*\\$-algebras introduced in our joint work with Winter. In addition to obtaining characterisations in the spirit of Dixmier’s work on central sequences in II\\$\_1\\$ factors, we establish the equivalence of uniform property \\$\\Gamma \\$, a suitable uniform version of McDuff’s property for \\$C^*\\$-algebras, and the existence of complemented partitions of unity for separable nuclear \\$C^*\\$-algebras with no finite dimensional representations and a compact (non-empty) tracial state space. As a consequence, for \\$C^*\\$-algebras as in the Toms–Winter conjecture, the combination of strict comparison and uniform property \\$\\Gamma \\$ is equivalent to Jiang–Su stability. We also show how these ideas can be combined with those of Matui–Sato to streamline Winter’s classification by embeddings technique.}", + issn = {1073-7928}, + doi = {10.1093/imrn/rnaa282}, + url = {https://doi.org/10.1093/imrn/rnaa282}, + eprint = {https://academic.oup.com/imrn/article-pdf/2022/13/9864/44360105/rnaa282.pdf}, +} + +@article {CETW22, + AUTHOR = {Castillejos, Jorge and Evington, Samuel and Tikuisis, Aaron + and White, Stuart}, + TITLE = {Uniform property {$\Gamma$}}, + JOURNAL = {Int. Math. Res. Not. IMRN}, + FJOURNAL = {International Mathematics Research Notices. IMRN}, + YEAR = {2022}, + NUMBER = {13}, + PAGES = {9864--9908}, + ISSN = {1073-7928}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {4447140}, + DOI = {10.1093/imrn/rnaa282}, + URL = {https://doi.org/10.1093/imrn/rnaa282}, +} + +# Chand-Robert +@article {ChandRobert23, + AUTHOR = {Chand, Abhinav and Robert, Leonel}, + TITLE = {Simplicity, bounded normal generation, and automatic + continuity of groups of unitaries}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {415}, + YEAR = {2023}, + PAGES = {Paper No. 108894, 52}, + ISSN = {0001-8708}, + MRCLASS = {46L05 (22E65 46H40)}, + MRNUMBER = {4543451}, + DOI = {10.1016/j.aim.2023.108894}, + URL = {https://doi.org/10.1016/j.aim.2023.108894}, +} + +# Chen +@book{Chen94Thesis, + AUTHOR = {Chen, Jianduan}, + TITLE = {Connes' invariant $\chi(\mathcal{M})$ and cohomology of groups}, + NOTE = {PhD thesis, University of California, Berkley}, + YEAR = {1994}, + MRCLASS = {Thesis} +} + +# Choda +@article {Choda70, + AUTHOR = {Choda, Hisashi}, + TITLE = {An extremal property of the polar decomposition in von + {N}eumann algebras}, + JOURNAL = {Proc. Japan Acad.}, + FJOURNAL = {Proceedings of the Japan Academy}, + VOLUME = {46}, + YEAR = {1970}, + PAGES = {341--344}, + ISSN = {0021-4280}, + MRCLASS = {46L10}, + MRNUMBER = {355624}, +MRREVIEWER = {Arthur\ Lieberman}, + URL = {http://projecteuclid.org/euclid.pja/1195520348}, +} + + +# Choi-Effros +@article {ChoiEffros76, + AUTHOR = {Choi, Man Duen and Effros, Edward G.}, + TITLE = {The completely positive lifting problem for + {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {104}, + YEAR = {1976}, + NUMBER = {3}, + PAGES = {585--609}, + ISSN = {0003-486X}, + MRCLASS = {46L05}, + MRNUMBER = {417795}, +MRREVIEWER = {Maurice J. Dupr\'{e}}, + DOI = {10.2307/1970968}, + URL = {https://doi.org/10.2307/1970968}, +} + +# Choi-Farah-Ozawa +@article {ChoiFarahOzawa14, + AUTHOR = {Choi, Yemon and Farah, Ilijas and Ozawa, Narutaka}, + TITLE = {A nonseparable amenable operator algebra which is not + isomorphic to a {C}*-algebra}, + JOURNAL = {Forum Math. Sigma}, + FJOURNAL = {Forum of Mathematics. Sigma}, + VOLUME = {2}, + YEAR = {2014}, + PAGES = {Paper No. e2, 12}, + ISSN = {2050-5094}, + MRCLASS = {47L30 (03E75 46L05)}, + MRNUMBER = {3177805}, + DOI = {10.1017/fms.2013.6}, + URL = {https://doi.org/10.1017/fms.2013.6}, +} + + +@article {MR3177805, + AUTHOR = {Choi, Yemon and Farah, Ilijas and Ozawa, Narutaka}, + TITLE = {A nonseparable amenable operator algebra which is not + isomorphic to a {${\rm C}^*$}-algebra}, + JOURNAL = {Forum Math. Sigma}, + FJOURNAL = {Forum of Mathematics. Sigma}, + VOLUME = {2}, + YEAR = {2014}, + PAGES = {Paper No. e2, 12}, + ISSN = {2050-5094}, + MRCLASS = {47L30 (03E75 46L05)}, + MRNUMBER = {3177805}, + DOI = {10.1017/fms.2013.6}, + URL = {https://doi.org/10.1017/fms.2013.6}, +} + + +# Connes +@article{Connes74almost, + title={Almost periodic states and factors of type {III}$_1$}, + author={Connes, Alain}, + journal={Journal of Functional Analysis}, + volume={16}, + number={4}, + pages={415--445}, + year={1974}, + publisher={Elsevier} +} + +@article {Connes75sur, + AUTHOR = {Connes, Alain}, + TITLE = {Sur la classification des facteurs de type {${\rm II}$}}, + JOURNAL = {C. R. Acad. Sci. Paris S\'er. A-B}, + FJOURNAL = {Comptes Rendus Hebdomadaires des S\'eances de l'Acad\'emie des + Sciences. S\'eries A et B}, + VOLUME = {281}, + YEAR = {1975}, + NUMBER = {1}, + PAGES = {Aii, A13--A15}, + ISSN = {0151-0509}, + MRCLASS = {46L10}, + MRNUMBER = {377534}, +MRREVIEWER = {S.\ Sakai}, +} + +@article {Connes75auto, + AUTHOR = {Connes, Alain}, + TITLE = {Classification of automorphisms of hyperfinite factors of type {II}$_1$ and {II}$_{\infty}$ and application to type {III} factors}, + JOURNAL = {Bull. Amer. Math. Soc.}, + FJOURNAL = {Bulletin of the American Mathematical Society}, + VOLUME = {81}, + YEAR = {1975}, + NUMBER = {6}, + PAGES = {1090--1092}, + ISSN = {0002-9904}, + MRCLASS = {46L10}, + MRNUMBER = {388117}, +MRREVIEWER = {Hisashi Choda}, + DOI = {10.1090/S0002-9904-1975-13929-2}, + URL = {https://doi.org/10.1090/S0002-9904-1975-13929-2}, +} + +@article {Connes75anti, + AUTHOR = {Connes, Alain}, + TITLE = {A factor not anti-isomorphic to itself}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {101}, + YEAR = {1975}, + PAGES = {536--554}, + ISSN = {0003-486X}, + MRCLASS = {46L10}, + MRNUMBER = {370209}, +MRREVIEWER = {E. St{\o}rmer}, + DOI = {10.2307/1970940}, + URL = {https://doi.org/10.2307/1970940}, +} + +@article {Connes75Outer, + AUTHOR = {Connes, Alain}, + TITLE = {Outer conjugacy classes of automorphisms of factors}, + JOURNAL = {Ann. Sci. \'{E}cole Norm. Sup. (4)}, + FJOURNAL = {Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure. + Quatri\`eme S\'{e}rie}, + VOLUME = {8}, + YEAR = {1975}, + NUMBER = {3}, + PAGES = {383--419}, + ISSN = {0012-9593}, + MRCLASS = {46L10}, + MRNUMBER = {394228}, +MRREVIEWER = {Hisashi\ Choda}, + URL = {http://www.numdam.org/item?id=ASENS_1975_4_8_3_383_0}, +} + +@article {Connes76, + AUTHOR = {Connes, Alain}, + TITLE = {Classification of injective factors Cases {II}$_1$, {II}$_\infty$, {III}$_\lambda$, $\lambda \neq 1$}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {104}, + YEAR = {1976}, + NUMBER = {1}, + PAGES = {73--115}, + ISSN = {0003-486X}, + MRCLASS = {46L10}, + MRNUMBER = {454659}, +MRREVIEWER = {Fran\c{c}ois Combes}, + DOI = {10.2307/1971057}, + URL = {https://doi.org/10.2307/1971057}, +} + + +@article {Connes77, + AUTHOR = {Connes, Alain}, + TITLE = {Periodic automorphisms of the hyperfinite factor of type {II}$_1$}, + JOURNAL = {Acta Sci. Math. (Szeged)}, + FJOURNAL = {Acta Universitatis Szegediensis. Acta Scientiarum + Mathematicarum}, + VOLUME = {39}, + YEAR = {1977}, + NUMBER = {1-2}, + PAGES = {39--66}, + ISSN = {0001-6969}, + MRCLASS = {46L10}, + MRNUMBER = {448101}, +MRREVIEWER = {Yoshinori Haga}, +} + +@article {Connes78, + AUTHOR = {Connes, Alain}, + TITLE = {On the cohomology of operator algebras}, + JOURNAL = {J. Functional Analysis}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {28}, + YEAR = {1978}, + NUMBER = {2}, + PAGES = {248--253}, + ISSN = {0022-1236}, + MRCLASS = {46L10}, + MRNUMBER = {493383}, +MRREVIEWER = {Man-Duen\ Choi}, + DOI = {10.1016/0022-1236(78)90088-5}, + URL = {https://doi.org/10.1016/0022-1236(78)90088-5}, +} + +@article {Connes85, + AUTHOR = {Connes, Alain}, + TITLE = {FACTORS OF TYPE {III}$_1$, PROPERTY {L}$_\lambda'$ AND CLOSURE OF INNER AUTOMORPHISMS}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {14}, + YEAR = {1985}, + NUMBER = {1}, + PAGES = {189--211}, + ISSN = {0379-4024}, + MRCLASS = {46L35}, + MRNUMBER = {789385}, +MRREVIEWER = {M. Takesaki}, +} + +# Conway +@book {Conway19, + AUTHOR = {Conway, John B.}, + TITLE = {A course in functional analysis}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {96}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1985}, + PAGES = {xiv+404}, + ISBN = {0-387-96042-2}, + MRCLASS = {46-01 (47-01)}, + MRNUMBER = {768926}, +MRREVIEWER = {Greg Robel}, + DOI = {10.1007/978-1-4757-3828-5}, + URL = {https://doi.org/10.1007/978-1-4757-3828-5}, +} + +# Cuntz +@article {Cuntz77, + AUTHOR = {Cuntz, Joachim}, + TITLE = {Simple {C}*-algebras generated by isometries}, + JOURNAL = {Comm. Math. Phys.}, + FJOURNAL = {Communications in Mathematical Physics}, + VOLUME = {57}, + YEAR = {1977}, + NUMBER = {2}, + PAGES = {173--185}, + ISSN = {0010-3616}, + MRCLASS = {46L05}, + MRNUMBER = {467330}, +MRREVIEWER = {E. St{\o}rmer}, + URL = {http://projecteuclid.org/euclid.cmp/1103901288}, +} + +@article {Cuntz81, + AUTHOR = {Cuntz, Joachim}, + TITLE = {{K}-theory for certain {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {113}, + YEAR = {1981}, + NUMBER = {1}, + PAGES = {181--197}, + ISSN = {0003-486X}, + MRCLASS = {46L05 (16A54 18G99 46M20 58G12)}, + MRNUMBER = {604046}, +MRREVIEWER = {Vern Paulsen}, + DOI = {10.2307/1971137}, + URL = {https://doi.org/10.2307/1971137}, +} + +# Cuntz-Pedersen +@article{CuntzPedersen79, + title={Equivalence and traces on {C}*-algebras}, + author={Cuntz, Joachim and Pedersen, Gert K.}, + journal={Journal of Functional Analysis}, + volume={33}, + number={2}, + pages={135--164}, + year={1979}, + publisher={Elsevier} +} +@article {CuntzPedersen79, + AUTHOR = {Cuntz, Joachim and Pedersen, Gert Kjaerg\.{a}rd}, + TITLE = {Equivalence and traces on {C}*-algebras}, + JOURNAL = {J. Functional Analysis}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {33}, + YEAR = {1979}, + NUMBER = {2}, + PAGES = {135--164}, + ISSN = {0022-1236}, + MRCLASS = {46L05}, + MRNUMBER = {546503}, +MRREVIEWER = {Richard I. Loebl}, + DOI = {10.1016/0022-1236(79)90108-3}, + URL = {https://doi.org/10.1016/0022-1236(79)90108-3}, +} + +### DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD +# Dadarlat +@article {Dadarlat95, + AUTHOR = {Dadarlat, Marius}, + TITLE = {Reduction to dimension three of local spectra of real rank + zero {C}*-algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {460}, + YEAR = {1995}, + PAGES = {189--212}, + ISSN = {0075-4102}, + MRCLASS = {46L85 (19K14 46L35)}, + MRNUMBER = {1316577}, +MRREVIEWER = {Terry A. Loring}, + DOI = {10.1515/crll.1995.460.189}, + URL = {https://doi.org/10.1515/crll.1995.460.189}, +} + +# Dadarlat-Loring +@article {DadarlatLoring96, + AUTHOR = {Dadarlat, Marius and Loring, Terry A.}, + TITLE = {A universal multicoefficient theorem for the {K}asparov + groups}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {84}, + YEAR = {1996}, + NUMBER = {2}, + PAGES = {355--377}, + ISSN = {0012-7094}, + MRCLASS = {46L80 (19K35 46L35)}, + MRNUMBER = {1404333}, +MRREVIEWER = {Kevin McClanahan}, + DOI = {10.1215/S0012-7094-96-08412-4}, + URL = {https://doi.org/10.1215/S0012-7094-96-08412-4}, +} + +@article {DadarlatWinter09, + AUTHOR = {Dadarlat, Marius and Winter, Wilhelm}, + TITLE = {On the {KK}-theory of strongly self-absorbing + {C}*-algebras}, + JOURNAL = {Math. Scand.}, + FJOURNAL = {Mathematica Scandinavica}, + VOLUME = {104}, + YEAR = {2009}, + NUMBER = {1}, + PAGES = {95--107}, + ISSN = {0025-5521}, + MRCLASS = {46L80 (19K35 46L05)}, + MRNUMBER = {2498373}, +MRREVIEWER = {Vladimir Manuilov}, + DOI = {10.7146/math.scand.a-15086}, + URL = {https://doi.org/10.7146/math.scand.a-15086}, +} + +# Davidson +@book {DavidsonBook1, + AUTHOR = {Davidson, Kenneth R.}, + TITLE = {{C}*-algebras by example}, + SERIES = {Fields Institute Monographs}, + VOLUME = {6}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {1996}, + PAGES = {xiv+309}, + ISBN = {0-8218-0599-1}, + MRCLASS = {46Lxx (46-01)}, + MRNUMBER = {1402012}, +MRREVIEWER = {Robert S. Doran}, + DOI = {10.1090/fim/006}, + URL = {https://doi.org/10.1090/fim/006}, +} + +# Davidson-Kennedy +@article{DavidsonKennedy19, + title={Noncommutative choquet theory}, + author={Davidson, Kenneth R and Kennedy, Matthew}, + journal={arXiv:1905.08436}, + year={2019} +} + + + +#de Lacerda Mortari +@book {deLacerdaMortari09, + AUTHOR = {Mortari, Fernando de Lacerda}, + TITLE = {Tracial state spaces of higher stable rank simple + {C}*-algebras}, + NOTE = {Thesis (Ph.D.)--University of Toronto (Canada)}, + PUBLISHER = {ProQuest LLC, Ann Arbor, MI}, + YEAR = {2009}, + PAGES = {41}, + ISBN = {978-0494-61035-0}, + MRCLASS = {Thesis}, + MRNUMBER = {2753146}, + URL = + {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:NR61035}, +} + +# de la Harpe +@incollection {dlHarpe79Moyen, + AUTHOR = {de la Harpe, Pierre}, + TITLE = {Moyennabilit\'{e} du groupe unitaire et propri\'{e}t\'{e} + {$P$} de {S}chwartz des alg\`ebres de von {N}eumann}, + BOOKTITLE = {Alg\`ebres d'op\'{e}rateurs ({S}\'{e}m., {L}es + {P}lans-sur-{B}ex, 1978)}, + SERIES = {Lecture Notes in Math.}, + VOLUME = {725}, + PAGES = {220--227}, + PUBLISHER = {Springer, Berlin}, + YEAR = {1979}, + ISBN = {3-540-09512-8}, + MRCLASS = {46L35 (22E65 58B25)}, + MRNUMBER = {548116}, +MRREVIEWER = {Marie\ Choda}, +} + +@article {dlHarpe79Simp, + AUTHOR = {de la Harpe, Pierre}, + TITLE = {Simplicity of the projective unitary groups defined by simple + factors}, + JOURNAL = {Comment. Math. Helv.}, + FJOURNAL = {Commentarii Mathematici Helvetici}, + VOLUME = {54}, + YEAR = {1979}, + NUMBER = {2}, + PAGES = {334--345}, + ISSN = {0010-2571,1420-8946}, + MRCLASS = {22E65 (46L99)}, + MRNUMBER = {535064}, +MRREVIEWER = {Raymond\ Streater}, + DOI = {10.1007/BF02566277}, + URL = {https://doi.org/10.1007/BF02566277}, +} + +@article {dlHarpe13, + AUTHOR = {de la Harpe, Pierre}, + TITLE = {Fuglede--{K}adison determinant: theme and variations}, + JOURNAL = {Proc. Natl. Acad. Sci. USA}, + FJOURNAL = {Proceedings of the National Academy of Sciences of the United + States of America}, + VOLUME = {110}, + YEAR = {2013}, + NUMBER = {40}, + PAGES = {15864--15877}, + ISSN = {0027-8424}, + MRCLASS = {46L10}, + MRNUMBER = {3363445}, + DOI = {10.1073/pnas.1202059110}, + URL = {https://doi.org/10.1073/pnas.1202059110}, +} + +# de la Harpe-McDuff +@article {dlHarpeMcDuff83, + AUTHOR = {de la Harpe, Pierre and McDuff, Dusa}, + TITLE = {Acyclic groups of automorphisms}, + JOURNAL = {Comment. Math. Helv.}, + FJOURNAL = {Commentarii Mathematici Helvetici}, + VOLUME = {58}, + YEAR = {1983}, + NUMBER = {1}, + PAGES = {48--71}, + ISSN = {0010-2571,1420-8946}, + MRCLASS = {22D45 (18F25 57S05)}, + MRNUMBER = {699006}, +MRREVIEWER = {A.\ A.\ Ranicki}, + DOI = {10.1007/BF02564624}, + URL = {https://doi.org/10.1007/BF02564624}, +} + +# de la Harpe-Skandalis +@article {dlHS84a, + AUTHOR = {de la Harpe, Pierre and Skandalis, Georges}, + TITLE = {D\'{e}terminant associ\'{e} \`a une trace sur une alg\'{e}bre de {B}anach}, + JOURNAL = {Ann. Inst. Fourier (Grenoble)}, + FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier}, + VOLUME = {34}, + YEAR = {1984}, + NUMBER = {1}, + PAGES = {241--260}, + ISSN = {0373-0956}, + MRCLASS = {46L80 (18F25 19K14 19K56 46L05 58G12)}, + MRNUMBER = {743629}, +MRREVIEWER = {G. A. Elliott}, + URL = {http://www.numdam.org/item?id=AIF_1984__34_1_241_0}, +} + +@article {dlHS84b, + AUTHOR = {de la Harpe, Pierre and Skandalis, Georges}, + TITLE = {Produits finis de commutateurs dans les {C}*-alg\`ebres}, + JOURNAL = {Ann. Inst. Fourier (Grenoble)}, + FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier}, + VOLUME = {34}, + YEAR = {1984}, + NUMBER = {4}, + PAGES = {169--202}, + ISSN = {0373-0956}, + MRCLASS = {46L05 (18F25 19B99 19C09 19K14 46L80 58G12)}, + MRNUMBER = {766279}, +MRREVIEWER = {G. A. Elliott}, + URL = {http://www.numdam.org/item?id=AIF_1984__34_4_169_0}, +} + +# Dummit-Foote +@book {DummitFoote, + AUTHOR = {Dummit, David S. and Foote, Richard M.}, + TITLE = {Abstract algebra}, + EDITION = {Third}, + PUBLISHER = {John Wiley \& Sons, Inc., Hoboken, NJ}, + YEAR = {2004}, + PAGES = {xii+932}, + ISBN = {0-471-43334-9}, + MRCLASS = {00-01 (16-01 20-01)}, + MRNUMBER = {2286236}, +} + +# Dye +@article {Dye53, + AUTHOR = {Dye, Henry A.}, + TITLE = {The unitary structure in finite rings of operators}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {20}, + YEAR = {1953}, + PAGES = {55--69}, + ISSN = {0012-7094}, + MRCLASS = {46.3X}, + MRNUMBER = {52695}, +MRREVIEWER = {F. I. Mautner}, + URL = {http://projecteuclid.org/euclid.dmj/1077465064}, +} + +@article {Dye55, + AUTHOR = {Dye, Henry A.}, + TITLE = {On the geometry of projections in certain operator algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {61}, + YEAR = {1955}, + PAGES = {73--89}, + ISSN = {0003-486X}, + MRCLASS = {46.0X}, + MRNUMBER = {66568}, +MRREVIEWER = {J. Dixmier}, + DOI = {10.2307/1969620}, + URL = {https://doi.org/10.2307/1969620}, +} + +# Dykema-Haagerup-Rordam +@article {DykemaHaagerupRordam97, + AUTHOR = {Dykema, Kenneth J. and Haagerup, Uffe and R{\o}rdam, Mikael}, + TITLE = {The stable rank of some free product {C}*-algebras}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {90}, + YEAR = {1997}, + NUMBER = {1}, + PAGES = {95--121}, + ISSN = {0012-7094,1547-7398}, + MRCLASS = {46L05 (19B10 19K56 46L35 46L80)}, + MRNUMBER = {1478545}, +MRREVIEWER = {Gustavo\ Corach}, + DOI = {10.1215/S0012-7094-97-09004-9}, + URL = {https://doi.org/10.1215/S0012-7094-97-09004-9}, +} + +# Dykema-Rordam +@article {DykemaRordam98, + AUTHOR = {Dykema, Kenneth J. and R{\o}rdam, Mikael}, + TITLE = {Projections in free product {C}*-algebras}, + JOURNAL = {Geom. Funct. Anal.}, + FJOURNAL = {Geometric and Functional Analysis}, + VOLUME = {8}, + YEAR = {1998}, + NUMBER = {1}, + PAGES = {1--16}, + ISSN = {1016-443X,1420-8970}, + MRCLASS = {46L05}, + MRNUMBER = {1601917}, +MRREVIEWER = {Gustavo\ Corach}, + DOI = {10.1007/s000390050046}, + URL = {https://doi.org/10.1007/s000390050046}, +} + +@article {DykemaRordam00, + AUTHOR = {Dykema, Kenneth J. and R{\o}rdam, Mikael}, + TITLE = {Projections in free product {C}*-algebras. {II}}, + JOURNAL = {Math. Z.}, + FJOURNAL = {Mathematische Zeitschrift}, + VOLUME = {234}, + YEAR = {2000}, + NUMBER = {1}, + PAGES = {103--113}, + ISSN = {0025-5874,1432-1823}, + MRCLASS = {46L09}, + MRNUMBER = {1759493}, +MRREVIEWER = {Gustavo\ Corach}, + DOI = {10.1007/s002090050505}, + URL = {https://doi.org/10.1007/s002090050505}, +} + + +### EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE +# Effros-Lance +@article {EffrosLance77, + AUTHOR = {Effros, Edward G. and Lance, E. Christopher}, + TITLE = {Tensor products of operator algebras}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {25}, + YEAR = {1977}, + NUMBER = {1}, + PAGES = {1--34}, + ISSN = {0001-8708,1090-2082}, + MRCLASS = {46L05 (46L10 46M05)}, + MRNUMBER = {448092}, +MRREVIEWER = {Jun\ Tomiyama}, + DOI = {10.1016/0001-8708(77)90085-8}, + URL = {https://doi.org/10.1016/0001-8708(77)90085-8}, +} + + +# Elliott +@article {Elliott76, + AUTHOR = {Elliott, George A.}, + TITLE = {On the classification of inductive limits of sequences of + semisimple finite-dimensional algebras}, + JOURNAL = {J. Algebra}, + FJOURNAL = {Journal of Algebra}, + VOLUME = {38}, + YEAR = {1976}, + NUMBER = {1}, + PAGES = {29--44}, + ISSN = {0021-8693}, + MRCLASS = {46L05 (16A46)}, + MRNUMBER = {397420}, +MRREVIEWER = {Horst Behncke}, + DOI = {10.1016/0021-8693(76)90242-8}, + URL = {https://doi.org/10.1016/0021-8693(76)90242-8}, +} + + +@incollection {Elliott93a, + AUTHOR = {Elliott, George A.}, + TITLE = {A classification of certain simple {$C^*$}-algebras}, + BOOKTITLE = {Quantum and non-commutative analysis ({K}yoto, 1992)}, + SERIES = {Math. Phys. Stud.}, + VOLUME = {16}, + PAGES = {373--385}, + PUBLISHER = {Kluwer Acad. Publ., Dordrecht}, + YEAR = {1993}, + ISBN = {0-7923-2532-X}, + MRCLASS = {46L35 (19K14 46L05 46L80)}, + MRNUMBER = {1276305}, +MRREVIEWER = {Terry\ A.\ Loring}, +} + +@article {Elliott93, + AUTHOR = {Elliott, George A.}, + TITLE = {On the classification of {C}*-algebras of real rank zero}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {443}, + YEAR = {1993}, + PAGES = {179--219}, + ISSN = {0075-4102}, + MRCLASS = {46L05 (19K14 46L80)}, + MRNUMBER = {1241132}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1515/crll.1993.443.179}, + URL = {https://doi.org/10.1515/crll.1993.443.179}, +} + +@article{Elliott22, +title="K-Theory and Traces", +author="Elliott, George A.", +journal="C. R. Math. Rep. Acad. Sci. Canada", +volume="44", +number="1", +pages="1--15", +year="2022" +} + +# Elliott-Gong-Lin-Niu +@article{EGLN15, + title={On the classification of simple amenable {C}*-algebras with finite decomposition rank, {II}}, + author={Elliott, George A. and Gong, Guihua and Lin, H. and Niu, Zhuang}, + journal={arXiv:1507.03437}, + year={2015} +} + + +@misc{ElliottLiNiu23, + title={Remarks on {V}illadsen algebras}, + author={Elliott, George A. and Li, Chun G. and Niu, Zhuang}, + year={2023}, + eprint={2209.10649}, + archivePrefix={arXiv}, + primaryClass={math.OA}, + NOTE={preprint} +} + +# Elliott-Niu +@article {ElliottNiu12, + AUTHOR = {Elliott, George A. and Niu, Zhuang}, + TITLE = {Extended rotation algebras: adjoining spectral projections to + rotation algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {665}, + YEAR = {2012}, + PAGES = {1--71}, + ISSN = {0075-4102}, + MRCLASS = {46L05}, + MRNUMBER = {2908740}, +MRREVIEWER = {William Paschke}, + DOI = {10.1515/CRELLE.2011.112}, + URL = {https://doi.org/10.1515/CRELLE.2011.112}, +} + +@article {ElliottNiu15, + AUTHOR = {Elliott, George A. and Niu, Zhuang}, + TITLE = {All irrational extended rotation algebras are {AF} algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {67}, + YEAR = {2015}, + NUMBER = {4}, + PAGES = {810--826}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {3361014}, +MRREVIEWER = {Katsutoshi Kawashima}, + DOI = {10.4153/CJM-2014-022-5}, + URL = {https://doi.org/10.4153/CJM-2014-022-5}, +} + +@incollection {ElliottNiu16, + AUTHOR = {Elliott, George A. and Niu, Zhuang}, + TITLE = {On the classification of simple amenable {C}*-algebras with + finite decomposition rank}, + BOOKTITLE = {Operator algebras and their applications}, + SERIES = {Contemp. Math.}, + VOLUME = {671}, + PAGES = {117--125}, + PUBLISHER = {Amer. Math. Soc., Providence, RI}, + YEAR = {2016}, + MRCLASS = {46L35}, + MRNUMBER = {3546681}, + DOI = {10.1090/conm/671/13506}, + URL = {https://doi.org/10.1090/conm/671/13506}, +} + +# Elliott-Villadsen +@article {ElliottVilladsen00, + AUTHOR = {Elliott, George A. and Villadsen, Jesper}, + TITLE = {Perforated ordered {$K_0$}-groups}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {52}, + YEAR = {2000}, + NUMBER = {6}, + PAGES = {1164--1191}, + ISSN = {0008-414X}, + MRCLASS = {46L80 (19K14 46L05)}, + MRNUMBER = {1794301}, + DOI = {10.4153/CJM-2000-049-9}, + URL = {https://doi.org/10.4153/CJM-2000-049-9}, +} + +# Echterhoff-Rordam +@article{EchterhoffRordam21, + title={Inclusions of {C}*-algebras arising from fixed-point algebras}, + author={Echterhoff, Siegfried and R{\o}rdam, Mikael}, + journal={arXiv:2108.08832}, + year={2021} +} + +# Enders-Schemaitat-Tikuisis +@article{EndersSchemaitatTikuisis23, + title={Corrigendum to ``{$K$}-theoretic characterization of {C}*-algebras with approximately inner flip''}, + author={Enders, Dominic and Schemaitat, Andr{\'e} and Tikuisis, Aaron}, + journal={arXiv:2303.11106}, + year={2023} +} + +# Evans-Kawahigashi +@book {EvansKawahigashiBook, + AUTHOR = {Evans, David E. and Kawahigashi, Yasuyuki}, + TITLE = {Quantum symmetries on operator algebras}, + SERIES = {Oxford Mathematical Monographs}, + NOTE = {Oxford Science Publications}, + PUBLISHER = {The Clarendon Press, Oxford University Press, New York}, + YEAR = {1998}, + PAGES = {xvi+829}, + ISBN = {0-19-851175-2}, + MRCLASS = {46L37 (19K99 46L60 46L80 46N55 82B10)}, + MRNUMBER = {1642584}, +MRREVIEWER = {Carl\ Winsl\o w}, +} + +# Evington-Giron Pacheco-Jones +@misc{EvingtonGiPachJones24, + title={Equivariant $\mathcal{D}$-stability for Actions of Tensor Categories}, + author={Samuel Evington and Sergio Girón Pacheco and Corey Jones}, + year={2024}, + eprint={2401.14238}, + archivePrefix={arXiv}, + primaryClass={math.OA}, + url={https://arxiv.org/abs/2401.14238}, +} + +# Evington-Pennig + +@article {EvingtonPennig16, + AUTHOR = {Evington, Samuel and Pennig, Ulrich}, + TITLE = {Locally trivial {W}*-bundles}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {27}, + YEAR = {2016}, + NUMBER = {11}, + PAGES = {1650088, 25}, + ISSN = {0129-167X,1793-6519}, + MRCLASS = {46L35}, + MRNUMBER = {3570373}, +MRREVIEWER = {Maria\ Grazia\ Viola}, + DOI = {10.1142/S0129167X16500889}, + URL = {https://doi.org/10.1142/S0129167X16500889}, +} + +### FFFFFFFFFFFFFFFFFFFFFF + + # Fack-de la Harpe +@article {FackdlHarpe80, + AUTHOR = {Fack, Thierry and de la Harpe, Pierre}, + TITLE = {Sommes de commutateurs dans les alg\`ebres de von {N}eumann + finies continues}, + JOURNAL = {Ann. Inst. Fourier (Grenoble)}, + FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier}, + VOLUME = {30}, + YEAR = {1980}, + NUMBER = {3}, + PAGES = {49--73}, + ISSN = {0373-0956,1777-5310}, + MRCLASS = {46L10}, + MRNUMBER = {597017}, +MRREVIEWER = {H.\ Halpern}, + URL = {http://www.numdam.org/item?id=AIF_1980__30_3_49_0}, +} + +#FangGeLi +@article {FangGeLi06, + AUTHOR = {Fang, Junsheng and Ge, Liming and Li, Weihua}, + TITLE = {Central sequence algebras of von {N}eumann algebras}, + JOURNAL = {Taiwanese J. Math.}, + FJOURNAL = {Taiwanese Journal of Mathematics}, + VOLUME = {10}, + YEAR = {2006}, + NUMBER = {1}, + PAGES = {187--200}, + ISSN = {1027-5487}, + MRCLASS = {46L10 (46M07)}, + MRNUMBER = {2186173}, +MRREVIEWER = {H. Halpern}, + DOI = {10.11650/twjm/1500403810}, + URL = {https://doi.org/10.11650/twjm/1500403810}, +} + +# Farah-Hirshberg +@article{FarahHirshberg16, + title={A simple {AF} algebra not isomorphic to its opposite}, + author={Farah, Ilijas and Hirshberg, Ilan}, + journal={arXiv:1612.01170}, + year={2016} +} + +@article {FarahHirshberg17, + AUTHOR = {Farah, Ilijas and Hirshberg, Ilan}, + TITLE = {Simple nuclear {C}*-algebras not isomorphic to their + opposites}, + JOURNAL = {Proc. Natl. Acad. Sci. USA}, + FJOURNAL = {Proceedings of the National Academy of Sciences of the United + States of America}, + VOLUME = {114}, + YEAR = {2017}, + NUMBER = {24}, + PAGES = {6244--6249}, + ISSN = {0027-8424}, + MRCLASS = {46L10}, + MRNUMBER = {3667529}, +MRREVIEWER = {Fyodor A. Sukochev}, + DOI = {10.1073/pnas.1619936114}, + URL = {https://doi.org/10.1073/pnas.1619936114}, +} + +# Fogang Takoutsing-Robert +@article{FoTakRobert24, + title={Distinguishing {C}*-algebras by their unitary groups}, + author={Fogang Takoutsing, Lionel and Robert, Leonel}, + journal={Proceedings of the American Mathematical Society}, + volume={152}, + number={10}, + pages={4301--4310}, + year={2024} +} + +# Folland +@book {Follandbook16, + AUTHOR = {Folland, Gerald B.}, + TITLE = {A course in abstract harmonic analysis}, + SERIES = {Textbooks in Mathematics}, + EDITION = {Second}, + PUBLISHER = {CRC Press, Boca Raton, FL}, + YEAR = {2016}, + PAGES = {xiii+305 pp.+loose errata}, + ISBN = {978-1-4987-2713-6}, + MRCLASS = {43-01 (22-01 42-01 46-01)}, + MRNUMBER = {3444405}, +MRREVIEWER = {D. L. Salinger}, +} + +# Fuglede-Kadison +@article {FugledeKadison52, + AUTHOR = {Fuglede, Bent and Kadison, Richard V.}, + TITLE = {Determinant theory in finite factors}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {55}, + YEAR = {1952}, + PAGES = {520--530}, + ISSN = {0003-486X}, + MRCLASS = {46.3X}, + MRNUMBER = {52696}, +MRREVIEWER = {F.\ I.\ Mautner}, + DOI = {10.2307/1969645}, + URL = {https://doi.org/10.2307/1969645}, +} + +### GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG +# Gabe +@article{Gabe19, + title={Classification of $\mathcal{O}_\infty$-stable {C}*-algebras}, + author={Gabe, James}, + journal={arXiv:1910.06504}, + year={2019} +} + +@article {Gabe20, + AUTHOR = {Gabe, James}, + TITLE = {A new proof of {K}irchberg's $\mathcal{O}_2$-stable classification}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {761}, + YEAR = {2020}, + PAGES = {247--289}, + ISSN = {0075-4102}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {4080250}, +MRREVIEWER = {Francesc Perera}, + DOI = {10.1515/crelle-2018-0010}, + URL = {https://doi.org/10.1515/crelle-2018-0010}, +} + +# Ganea +@article {Ganea68, + AUTHOR = {Ganea, Tudor}, + TITLE = {Homologie et extensions centrales de groupes}, + JOURNAL = {C. R. Acad. Sci. Paris S\'{e}r. A-B}, + FJOURNAL = {Comptes Rendus Hebdomadaires des S\'{e}ances de l'Acad\'{e}mie + des Sciences. S\'{e}ries A et B}, + VOLUME = {266}, + YEAR = {1968}, + PAGES = {A556--A558}, + ISSN = {0151-0509}, + MRCLASS = {20.48 (18.00)}, + MRNUMBER = {231914}, +MRREVIEWER = {U.\ Stammbach}, +} + +# Gardella +@article{Gardella17, + title={{R}okhlin-type properties for group actions on {C}*-algebras}, + author={Gardella, Eusebio}, + journal={Lecture Notes, IPM, Tehran}, + year={2017} +} + +@book {GardellaBook, + AUTHOR = {Gardella, Emilio Eusebio}, + TITLE = {Compact group actions on {C}*-algebras: classification, + non-classifiability, and crossed products and rigidity results + for {L}p-operator algebras}, + NOTE = {Thesis (Ph.D.)--University of Oregon}, + PUBLISHER = {ProQuest LLC, Ann Arbor, MI}, + YEAR = {2015}, + PAGES = {713}, + ISBN = {978-1321-96796-8}, + MRCLASS = {Thesis}, + MRNUMBER = {3407494}, + URL = + {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3717348}, +} + +# Gardella-Hirshberg +@article{GardellaHirshberg18, + title={Strongly outer actions of amenable groups on $\mathcal{Z}$-stable {C}*-algebras}, + author={Gardella, Eusebio and Hirshberg, Ilan}, + journal={arXiv:1811.00447}, + year={2018} +} + +# Gardella-Lupini +@article {GardellaLupini18, + AUTHOR = {Gardella, Eusebio and Lupini, Martino}, + TITLE = {Applications of model theory to {C}*-dynamics}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {275}, + YEAR = {2018}, + NUMBER = {7}, + PAGES = {1889--1942}, + ISSN = {0022-1236}, + MRCLASS = {03C98 (28D05 37A55 46L40 46L55 46M07)}, + MRNUMBER = {3832010}, +MRREVIEWER = {Alessandro Vignati}, + DOI = {10.1016/j.jfa.2018.03.020}, + URL = {https://doi.org/10.1016/j.jfa.2018.03.020}, +} + +# Gardella-Geffen-Naryshkin-Vaccaro +@article{GardellaGeffenNaryshkinVaccaro24, + title={Dynamical comparison and Z-stability for crossed products of simple C⁎-algebras}, + author={Gardella, Eusebio and Geffen, Shirly and Naryshkin, Petr and Vaccaro, Andrea}, + journal={Advances in Mathematics}, + volume={438}, + pages={109471}, + year={2024}, + publisher={Elsevier} +} + +# Gardella-Perera +@article{GardellaPerera22, + title={The modern theory of Cuntz semigroups of C*-algebras}, + author={Gardella, Eusebio and Perera, Francesc}, + journal={arXiv preprint arXiv:2212.02290}, + year={2022} +} + + +# Ge +@article {Ge95, + AUTHOR = {Ge, Liming}, + TITLE = {A splitting property for subalgebras of tensor products}, + JOURNAL = {Bull. Amer. Math. Soc. (N.S.)}, + FJOURNAL = {American Mathematical Society. Bulletin. New Series}, + VOLUME = {32}, + YEAR = {1995}, + NUMBER = {1}, + PAGES = {57--60}, + ISSN = {0273-0979,1088-9485}, + MRCLASS = {46L35 (46L10 46M05)}, + MRNUMBER = {1273397}, +MRREVIEWER = {Sze-Kai\ Tsui}, + DOI = {10.1090/S0273-0979-1995-00556-2}, + URL = {https://doi.org/10.1090/S0273-0979-1995-00556-2}, +} + +@article {GeKadison96, + AUTHOR = {Ge, Liming and Kadison, Richard V.}, + TITLE = {On tensor products for von {N}eumann algebras}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {123}, + YEAR = {1996}, + NUMBER = {3}, + PAGES = {453--466}, + ISSN = {0020-9910,1432-1297}, + MRCLASS = {46L10 (46M05)}, + MRNUMBER = {1383957}, +MRREVIEWER = {Sze-Kai\ Tsui}, + DOI = {10.1007/s002220050036}, + URL = {https://doi.org/10.1007/s002220050036}, +} + +# Geffen-Ursu +@article{GeffenUrsu23, + title={Simplicity of crossed products by FC-hypercentral groups}, + author={Geffen, Shirly and Ursu, Dan}, + journal={arXiv:2304.07852}, + year={2023} +} + +# Germain +@article{Germain96, + author = {Emmanuel Germain}, + title = {{KK}-theory of reduced free-product {C}*-algebras}, + volume = {82}, + journal = {Duke Mathematical Journal}, + number = {3}, + publisher = {Duke University Press}, + pages = {707 -- 723}, + year = {1996}, + doi = {10.1215/S0012-7094-96-08229-0}, + URL = {https://doi.org/10.1215/S0012-7094-96-08229-0} +} + +@article{Germain97, + url = {https://doi.org/10.1515/crll.1997.485.1}, + title = {{KK}-theory of the full free product of unital {C}*-algebras}, + author = {Emmanuel Germain}, + pages = {1--10}, + volume = {1997}, + number = {485}, + journal = {Journal f\"{u}r die reine und angewandte Mathematik}, + doi = {doi:10.1515/crll.1997.485.1}, + year = {1997}, + lastchecked = {2024-07-09} +} + +# Giordano-Pestov +@article {GiordanoPestov02, + AUTHOR = {Giordano, Thierry and Pestov, Vladimir}, + TITLE = {Some extremely amenable groups}, + JOURNAL = {C. R. Math. Acad. Sci. Paris}, + FJOURNAL = {Comptes Rendus Math\'{e}matique. Acad\'{e}mie des Sciences. + Paris}, + VOLUME = {334}, + YEAR = {2002}, + NUMBER = {4}, + PAGES = {273--278}, + ISSN = {1631-073X,1778-3569}, + MRCLASS = {43A07 (46L10)}, + MRNUMBER = {1891002}, + DOI = {10.1016/S1631-073X(02)02218-5}, + URL = {https/://doi.org/10.1016/S1631-073X(02)02218-5}, +} + +@article {GiordanoPestov07, + AUTHOR = {Giordano, Thierry and Pestov, Vladimir}, + TITLE = {Some extremely amenable groups related to operator algebras + and ergodic theory}, + JOURNAL = {J. Inst. Math. Jussieu}, + FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. + Journal de l'Institut de Math\'{e}matiques de Jussieu}, + VOLUME = {6}, + YEAR = {2007}, + NUMBER = {2}, + PAGES = {279--315}, + ISSN = {1474-7480,1475-3030}, + MRCLASS = {22D25 (22F50 37A15 43A07 46L99)}, + MRNUMBER = {2311665}, +MRREVIEWER = {Matthew\ D.\ Daws}, + DOI = {10.1017/S1474748006000090}, + URL = {https://doi.org/10.1017/S1474748006000090}, +} + + +# Giordano-Sierakowski +@article {GiordanoSierakowski16, + AUTHOR = {Giordano, Thierry and Sierakowski, Adam}, + TITLE = {The general linear group as a complete invariant for + {C}*-algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {76}, + YEAR = {2016}, + NUMBER = {2}, + PAGES = {249--269}, + ISSN = {0379-4024}, + MRCLASS = {46L35 (47L80)}, + MRNUMBER = {3552377}, +MRREVIEWER = {Marius V. Ionescu}, + DOI = {10.7900/jot.2015may27.2112}, + URL = {https://doi.org/10.7900/jot.2015may27.2112}, +} + +# Giordano-Skandalis +@article {GiordanoSkandalis85, + AUTHOR = {Giordano, T. and Skandalis, G.}, + TITLE = {{K}rieger factors isomorphic to their tensor square and pure + point spectrum flows}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {64}, + YEAR = {1985}, + NUMBER = {2}, + PAGES = {209--226}, + ISSN = {0022-1236}, + MRCLASS = {46L35}, + MRNUMBER = {812392}, +MRREVIEWER = {Michel Hilsum}, + DOI = {10.1016/0022-1236(85)90075-8}, + URL = {https://doi.org/10.1016/0022-1236(85)90075-8}, +} + +#Giron Pacheco-Neagu +@misc{GiPachNeagu24, + title={An Elliott intertwining approach to classifying actions of {C}*-tensor categories}, + author={Sergio Girón Pacheco and Robert Neagu}, + year={2024}, + eprint={2310.18125}, + archivePrefix={arXiv}, + primaryClass={math.OA}, + url={https://arxiv.org/abs/2310.18125}, +} + + +# Gong +@article {Gong97, + AUTHOR = {Gong, Guihua}, + TITLE = {On inductive limits of matrix algebras over higher-dimensional + spaces. {I}, {II}}, + JOURNAL = {Math. Scand.}, + FJOURNAL = {Mathematica Scandinavica}, + VOLUME = {80}, + YEAR = {1997}, + NUMBER = {1}, + PAGES = {41--55, 56--100}, + ISSN = {0025-5521}, + MRCLASS = {46L05 (19K14 46L35 46L80)}, + MRNUMBER = {1466905}, +MRREVIEWER = {Terry A. Loring}, + DOI = {10.7146/math.scand.a-12611}, + URL = {https://doi.org/10.7146/math.scand.a-12611}, +} + +# Gong-Jiang-Su +@article {GongJiangSu00, + AUTHOR = {Gong, Guihua and Jiang, Xinhui and Su, Hongbing}, + TITLE = {Obstructions to $\mathcal{Z}$-stability for unital simple + {$C^*$}-algebras}, + JOURNAL = {Canad. Math. Bull.}, + FJOURNAL = {Canadian Mathematical Bulletin. Bulletin Canadien de + Math\'{e}matiques}, + VOLUME = {43}, + YEAR = {2000}, + NUMBER = {4}, + PAGES = {418--426}, + ISSN = {0008-4395}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {1793944}, +MRREVIEWER = {Takahiro Sudo}, + DOI = {10.4153/CMB-2000-050-1}, + URL = {https://doi.org/10.4153/CMB-2000-050-1}, +} + +# Gong-Lin-Niu +@article {GongLinNiu20I, + AUTHOR = {Gong, Guihua and Lin, Huaxin and Niu, Zhuang}, + TITLE = {A classification of finite simple amenable $\mathcal{Z}$-stable + {C}*-algebras, {I}: {C}*-algebras with generalized + tracial rank one}, + JOURNAL = {C. R. Math. Acad. Sci. Soc. R. Can.}, + FJOURNAL = {Comptes Rendus Math\'{e}matiques de l'Acad\'{e}mie des Sciences. La + Soci\'{e}t\'{e} Royale du Canada. Mathematical Reports of the Academy + of Science. The Royal Society of Canada}, + VOLUME = {42}, + YEAR = {2020}, + NUMBER = {3}, + PAGES = {63--450}, + ISSN = {0706-1994}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {4215379}, +} + +@article {GongLinNiu20II, + AUTHOR = {Gong, Guihua and Lin, Huaxin and Niu, Zhuang}, + TITLE = {A classification of finite simple amenable $\mathcal{Z}$-stable {C}*-algebras, {II}: {C}*-algebras with rational generalized tracial rank one}, + JOURNAL = {C. R. Math. Acad. Sci. Soc. R. Can.}, + FJOURNAL = {Comptes Rendus Math\'{e}matiques de l'Acad\'{e}mie des Sciences. La + Soci\'{e}t\'{e} Royale du Canada. Mathematical Reports of the Academy + of Science. The Royal Society of Canada}, + VOLUME = {42}, + YEAR = {2020}, + NUMBER = {4}, + PAGES = {451--539}, + ISSN = {0706-1994}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {4215380}, +} + +# Gong-Lin-Xue +@article {GongLinXue15, + AUTHOR = {Gong, Guihua and Lin, Huaxin and Xue, Yifeng}, + TITLE = {Determinant rank of {$C^*$}-algebras}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {274}, + YEAR = {2015}, + NUMBER = {2}, + PAGES = {405--436}, + ISSN = {0030-8730}, + MRCLASS = {46L06 (46L35 46L80)}, + MRNUMBER = {3332910}, +MRREVIEWER = {Claire Anantharaman-Delaroche}, + DOI = {10.2140/pjm.2015.274.405}, + URL = {https://doi.org/10.2140/pjm.2015.274.405}, +} + +# Goodearl +@book {Goodearlbook, + AUTHOR = {Goodearl, Kenneth R.}, + TITLE = {Partially ordered abelian groups with interpolation}, + SERIES = {Mathematical Surveys and Monographs}, + VOLUME = {20}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {1986}, + PAGES = {xxii+336}, + ISBN = {0-8218-1520-2}, + MRCLASS = {06F20 (16A54 18F25 19K14 46A55 46L80)}, + MRNUMBER = {845783}, +MRREVIEWER = {G. A. Elliott}, + DOI = {10.1090/surv/020}, + URL = {https://doi.org/10.1090/surv/020}, +} + +# Guentner-Willett-Yu +@article{GuentnerWillettYu16, + title={Dynamical complexity and controlled operator {K}-theory}, + author={Guentner, Erik and Willett, Rufus and Yu, Guoliang}, + journal={arXiv:1609.02093}, + year={2016} +} + +### HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + +# Haagerup +@article {Haagerup79I, + AUTHOR = {Haagerup, Uffe}, + TITLE = {Operator-valued weights in von {N}eumann algebras. {I}}, + JOURNAL = {J. Functional Analysis}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {32}, + YEAR = {1979}, + NUMBER = {2}, + PAGES = {175--206}, + ISSN = {0022-1236}, + MRCLASS = {46L10 (46L50)}, + MRNUMBER = {534673}, +MRREVIEWER = {M.\ Takesaki}, + DOI = {10.1016/0022-1236(79)90053-3}, + URL = {https://doi.org/10.1016/0022-1236(79)90053-3}, +} + +@article {Haagerup79II, + AUTHOR = {Haagerup, Uffe}, + TITLE = {Operator-valued weights in von {N}eumann algebras. {II}}, + JOURNAL = {J. Functional Analysis}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {33}, + YEAR = {1979}, + NUMBER = {3}, + PAGES = {339--361}, + ISSN = {0022-1236}, + MRCLASS = {46L10 (46L50)}, + MRNUMBER = {549119}, +MRREVIEWER = {M.\ Takesaki}, + DOI = {10.1016/0022-1236(79)90072-7}, + URL = {https://doi.org/10.1016/0022-1236(79)90072-7}, +} + +@article {Haagerup83, + AUTHOR = {Haagerup, Uffe}, + TITLE = {All nuclear {C}*-algebras are amenable}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {74}, + YEAR = {1983}, + NUMBER = {2}, + PAGES = {305--319}, + ISSN = {0020-9910,1432-1297}, + MRCLASS = {46L05}, + MRNUMBER = {723220}, +MRREVIEWER = {J.\ W.\ Bunce}, + DOI = {10.1007/BF01394319}, + URL = {https://doi.org/10.1007/BF01394319}, +} + +@article {Haagerup87, + AUTHOR = {Haagerup, Uffe}, + TITLE = {{C}onnes' bicentralizer problem and uniqueness of the injective + factor of type {III}$_1$}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {158}, + YEAR = {1987}, + NUMBER = {1-2}, + PAGES = {95--148}, + ISSN = {0001-5962}, + MRCLASS = {46L35}, + MRNUMBER = {880070}, +MRREVIEWER = {Steve Wright}, + DOI = {10.1007/BF02392257}, + URL = {https://doi.org/10.1007/BF02392257}, +} + +# Hall +@book {HallBook, + AUTHOR = {Hall, Brian}, + TITLE = {Lie groups, {L}ie algebras, and representations}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {222}, + EDITION = {Second}, + NOTE = {An elementary introduction}, + PUBLISHER = {Springer, Cham}, + YEAR = {2015}, + PAGES = {xiv+449}, + ISBN = {978-3-319-13466-6; 978-3-319-13467-3}, + MRCLASS = {22-01 (17-01)}, + MRNUMBER = {3331229}, + DOI = {10.1007/978-3-319-13467-3}, + URL = {https://doi.org/10.1007/978-3-319-13467-3}, +} + +#Handelman +@article {Handelman78, + AUTHOR = {Handelman, David}, + TITLE = {{$K_0$} of von {N}eumann and {AF} {C}*-algebras}, + JOURNAL = {Quart. J. Math. Oxford Ser. (2)}, + FJOURNAL = {The Quarterly Journal of Mathematics. Oxford. Second Series}, + VOLUME = {29}, + YEAR = {1978}, + NUMBER = {116}, + PAGES = {427--441}, + ISSN = {0033-5606,1464-3847}, + MRCLASS = {46L05}, + MRNUMBER = {517736}, +MRREVIEWER = {Mi-Soo\ Smith}, + DOI = {10.1093/qmath/29.4.427}, + URL = {https://doi.org/10.1093/qmath/29.4.427}, +} + +@article {Handelman79, + AUTHOR = {Handelman, David}, + TITLE = {Stable range in {AW}* algebras}, + JOURNAL = {Proc. Amer. Math. Soc.}, + FJOURNAL = {Proceedings of the American Mathematical Society}, + VOLUME = {76}, + YEAR = {1979}, + NUMBER = {2}, + PAGES = {241--249}, + ISSN = {0002-9939,1088-6826}, + MRCLASS = {46L10 (16A45)}, + MRNUMBER = {537081}, +MRREVIEWER = {Jean-Marie\ Schwartz}, + DOI = {10.2307/2042996}, + URL = {https://doi.org/10.2307/2042996}, +} + +# Handelman-Rossmann +@article {HandelmanRossmann84, + AUTHOR = {Handelman, David and Rossmann, Wulf}, + TITLE = {Product type actions of finite and compact groups}, + JOURNAL = {Indiana Univ. Math. J.}, + FJOURNAL = {Indiana University Mathematics Journal}, + VOLUME = {33}, + YEAR = {1984}, + NUMBER = {4}, + PAGES = {479--509}, + ISSN = {0022-2518,1943-5258}, + MRCLASS = {46L80 (16A54 18F25 22D25 46L55 46M20)}, + MRNUMBER = {749311}, +MRREVIEWER = {Colin\ E.\ Sutherland}, + DOI = {10.1512/iumj.1984.33.33026}, + URL = {https://doi.org/10.1512/iumj.1984.33.33026}, +} + +@article {HandelmanRossmann85, + AUTHOR = {Handelman, David and Rossmann, Wulf}, + TITLE = {Actions of compact groups on {AF} {C}*-algebras}, + JOURNAL = {Illinois J. Math.}, + FJOURNAL = {Illinois Journal of Mathematics}, + VOLUME = {29}, + YEAR = {1985}, + NUMBER = {1}, + PAGES = {51--95}, + ISSN = {0019-2082,1945-6581}, + MRCLASS = {46L55 (22D25 46L80)}, + MRNUMBER = {769758}, +MRREVIEWER = {Thierry\ Fack}, + URL = {http://projecteuclid.org/euclid.ijm/1256045841}, +} + +# Hartman-Kalantar +@article {HartmanKalantar23, + AUTHOR = {Hartman, Yair and Kalantar, Mehrdad}, + TITLE = {Stationary {$C^*$}-dynamical systems}, + NOTE = {With an appendix by Uri Bader, Hartman and Kalantar}, + JOURNAL = {J. Eur. Math. Soc. (JEMS)}, + FJOURNAL = {Journal of the European Mathematical Society (JEMS)}, + VOLUME = {25}, + YEAR = {2023}, + NUMBER = {5}, + PAGES = {1783--1821}, + ISSN = {1435-9855,1435-9863}, + MRCLASS = {46L05 (20C07 37A55 46L35 46L55)}, + MRNUMBER = {4592860}, +MRREVIEWER = {Catalin\ Badea}, + DOI = {10.4171/jems/1225}, + URL = {https://doi.org/10.4171/jems/1225}, +} + + +# Hatcher +@book {HatcherAT, + AUTHOR = {Hatcher, Allen}, + TITLE = {Algebraic topology}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {2002}, + PAGES = {xii+544}, + ISBN = {0-521-79160-X; 0-521-79540-0}, + MRCLASS = {55-01 (55-00)}, + MRNUMBER = {1867354}, +MRREVIEWER = {Donald\ W.\ Kahn}, +} + +# Hatori-Molnar +@article {HatoriMolnar14, + AUTHOR = {Hatori, Osamu and Moln\'{a}r, Lajos}, + TITLE = {Isometries of the unitary groups and {T}hompson isometries of + the spaces of invertible positive elements in + {C}*-algebras}, + JOURNAL = {J. Math. Anal. Appl.}, + FJOURNAL = {Journal of Mathematical Analysis and Applications}, + VOLUME = {409}, + YEAR = {2014}, + NUMBER = {1}, + PAGES = {158--167}, + ISSN = {0022-247X}, + MRCLASS = {46L05}, + MRNUMBER = {3095026}, +MRREVIEWER = {Yangping Jing}, + DOI = {10.1016/j.jmaa.2013.06.065}, + URL = {https://doi.org/10.1016/j.jmaa.2013.06.065}, +} + +# Herman-Ocneanu +@article {HermanOcneanu84, + AUTHOR = {Herman, Richard H. and Ocneanu, Adrian}, + TITLE = {Stability for integer actions on {UHF} {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {59}, + YEAR = {1984}, + NUMBER = {1}, + PAGES = {132--144}, + ISSN = {0022-1236}, + MRCLASS = {46L40}, + MRNUMBER = {763780}, +MRREVIEWER = {Michel Hilsum}, + DOI = {10.1016/0022-1236(84)90056-9}, + URL = {https://doi.org/10.1016/0022-1236(84)90056-9}, +} + +# Higson +@article {Higson88, + AUTHOR = {Higson, Nigel}, + TITLE = {Algebraic {K}-theory of stable {C}*-algebras}, + JOURNAL = {Adv. in Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {67}, + YEAR = {1988}, + NUMBER = {1}, + PAGES = {140}, + ISSN = {0001-8708}, + MRCLASS = {46L80 (19K14 46M20)}, + MRNUMBER = {922140}, +MRREVIEWER = {Cornel Pasnicu}, + DOI = {10.1016/0001-8708(88)90034-5}, + URL = {https://doi.org/10.1016/0001-8708(88)90034-5}, +} + +# Hirshberg-Rordam-Winter +@article {HirshbergRordamWinter07, + AUTHOR = {Hirshberg, Ilan and R{\o}rdam, Mikael and Winter, Wilhelm}, + TITLE = {$\mathcal{C}_0({X})$-algebras, stability and strongly self-absorbing {C}*-algebras}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {339}, + YEAR = {2007}, + NUMBER = {3}, + PAGES = {695--732}, + ISSN = {0025-5831}, + MRCLASS = {46L05 (47L40)}, + MRNUMBER = {2336064}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.1007/s00208-007-0129-8}, + URL = {https://doi.org/10.1007/s00208-007-0129-8}, +} + +# Hirshberg-Orovitz +@article {HirshbergOrovitz13, + AUTHOR = {Hirshberg, Ilan and Orovitz, Joav}, + TITLE = {Tracially $\mathcal{Z}$-absorbing {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {265}, + YEAR = {2013}, + NUMBER = {5}, + PAGES = {765--785}, + ISSN = {0022-1236}, + MRCLASS = {46L05}, + MRNUMBER = {3063095}, +MRREVIEWER = {Stuart A. White}, + DOI = {10.1016/j.jfa.2013.05.005}, + URL = {https://doi.org/10.1016/j.jfa.2013.05.005}, +} + +# Hirshberg-Winter +@article {HirshbergWinter07, + AUTHOR = {Hirshberg, Ilan and Winter, Wilhelm}, + TITLE = {{R}okhlin actions and self-absorbing {C}*-algebras}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {233}, + YEAR = {2007}, + NUMBER = {1}, + PAGES = {125--143}, + ISSN = {0030-8730}, + MRCLASS = {46L05 (46L55)}, + MRNUMBER = {2366371}, +MRREVIEWER = {Efren Ruiz}, + DOI = {10.2140/pjm.2007.233.125}, + URL = {https://doi.org/10.2140/pjm.2007.233.125}, +} + +@article {HirshbergWinter08, + AUTHOR = {Hirshberg, Ilan and Winter, Wilhelm}, + TITLE = {Permutations of strongly self-absorbing {C}*-algebras}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {19}, + YEAR = {2008}, + NUMBER = {9}, + PAGES = {1137--1145}, + ISSN = {0129-167X}, + MRCLASS = {46L05 (46L45 46L55)}, + MRNUMBER = {2458564}, +MRREVIEWER = {Yoshikazu Katayama}, + DOI = {10.1142/S0129167X08005011}, + URL = {https://doi.org/10.1142/S0129167X08005011}, +} + +# Hoschs, Kaad, Schemaitat +@article {HochsKaadSchemaitat18, + AUTHOR = {Hochs, Peter and Kaad, Jens and Schemaitat, Andr\'{e}}, + TITLE = {Algebraic {$K$}-theory and a semifinite {F}uglede-{K}adison + determinant}, + JOURNAL = {Ann. K-Theory}, + FJOURNAL = {Annals of K-Theory}, + VOLUME = {3}, + YEAR = {2018}, + NUMBER = {2}, + PAGES = {193--206}, + ISSN = {2379-1683,2379-1691}, + MRCLASS = {46L80}, + MRNUMBER = {3781426}, +MRREVIEWER = {Christopher\ Jack\ Bourne}, + DOI = {10.2140/akt.2018.3.193}, + URL = {https://doi.org/10.2140/akt.2018.3.193}, +} + +# Husemoller +@book {Husemoller66, + AUTHOR = {Husemoller, Dale}, + TITLE = {Fibre bundles}, + PUBLISHER = {McGraw-Hill Book Co., New York-London-Sydney}, + YEAR = {1966}, + PAGES = {xiv+300}, + MRCLASS = {57.30 (55.00)}, + MRNUMBER = {0229247}, +MRREVIEWER = {R. Williamson}, +} + +# Hurewicz-Wallman +@book {HurewiczWallman, + AUTHOR = {Hurewicz, Witold and Wallman, Henry}, + TITLE = {Dimension theory}, + SERIES = {Princeton Mathematical Series, vol. 4}, + PUBLISHER = {Princeton University Press, Princeton, N. J.}, + YEAR = {1941}, + PAGES = {vii+165}, + MRCLASS = {56.0X}, + MRNUMBER = {0006493}, +MRREVIEWER = {H. Whitney}, +} + + +### IIIIIIIIIIIIIIIIIIIIIIIIIIIIII + +# Ioana +@article {Ioana07, + AUTHOR = {Ioana, Adrian}, + TITLE = {A relative version of {C}onnes' {$\chi(M)$} invariant and + existence of orbit inequivalent actions}, + JOURNAL = {Ergodic Theory Dynam. Systems}, + FJOURNAL = {Ergodic Theory and Dynamical Systems}, + VOLUME = {27}, + YEAR = {2007}, + NUMBER = {4}, + PAGES = {1199--1213}, + ISSN = {0143-3857,1469-4417}, + MRCLASS = {37A20 (28D15 46L10 46L55 47A15)}, + MRNUMBER = {2342972}, +MRREVIEWER = {Alain\ Valette}, + DOI = {10.1017/S0143385706000666}, + URL = {https://doi.org/10.1017/S0143385706000666}, +} + +# Ivanescu-Kucerovsky +@article{IvanescuKucerovsky23, + title={Villadsen algebras}, + author={Cristian Ivanescu and Dan Kucerovsky}, + year={2023}, + eprint={2306.03943}, + archivePrefix={arXiv}, + primaryClass={math.OA}, + NOTE={preprint}, +} + +# Izumi +@article{Izumi02, + title={Inclusions of simple {C}*-algebras}, + author={Izumi, Masaki}, + year={2002}, + volume = {547}, + pages = {97--138}, + journal = {J. Reine Angew. Math.}, + publisher={Walter de Gruyter GmbH \& Co. KG Berlin, Germany}, +} + +@article {Izumi04, + AUTHOR = {Izumi, Masaki}, + TITLE = {Finite group actions on {C}*-algebras with the {R}ohlin property. {I}}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {122}, + YEAR = {2004}, + NUMBER = {2}, + PAGES = {233--280}, + ISSN = {0012-7094}, + MRCLASS = {46L55 (19K35 46L35 46L40 46L80)}, + MRNUMBER = {2053753}, +MRREVIEWER = {Valentin Deaconu}, + DOI = {10.1215/S0012-7094-04-12221-3}, + URL = {https://doi.org/10.1215/S0012-7094-04-12221-3}, +} + +### JJJJJJJJJJJJJJJJJJJJJ +# Jiang +@article{Jiang97, + title={Nonstable {K}-theory for $\mathcal{Z}$-stable {C}*-algebras}, + author={Jiang, Xinhui}, + journal={arXiv preprint math/9707228}, + year={1997} +} + +# Jiang-Su +@article {JiangSu99, + AUTHOR = {Jiang, Xinhui and Su, Hongbing}, + TITLE = {On a simple unital projectionless {C}*-algebra}, + JOURNAL = {Amer. J. Math.}, + FJOURNAL = {American Journal of Mathematics}, + VOLUME = {121}, + YEAR = {1999}, + NUMBER = {2}, + PAGES = {359--413}, + ISSN = {0002-9327}, + MRCLASS = {46L35 (19K35 46L80)}, + MRNUMBER = {1680321}, +MRREVIEWER = {Vicumpriya S. Perera}, + URL = + {http://muse.jhu.edu/journals/american_journal_of_mathematics/v121/121.2jiang.pdf}, +} + +#Jolissaint +@article{Jolissaint16, + AUTHOR = {Jolissaint, Paul}, + TITLE = {Relative inner amenability and relative property gamma}, + JOURNAL = {Math. Scand.}, + FJOURNAL = {Mathematica Scandinavica}, + VOLUME = {119}, + YEAR = {2016}, + NUMBER = {2}, + PAGES = {293--319}, + ISSN = {0025-5521}, + MRCLASS = {22D25 (22D10 22F05 43A07 46L37)}, + MRNUMBER = {3570949}, +MRREVIEWER = {Jonathan M. Rosenberg}, + DOI = {10.7146/math.scand.a-24748}, + URL = {https://doi.org/10.7146/math.scand.a-24748}, +} + + +#Jones +@incollection {Jones79, + AUTHOR = {Jones, Vaughan F. R.}, + TITLE = {An invariant for group actions}, + BOOKTITLE = {Alg\`ebres d'op\'{e}rateurs ({S}\'{e}m., {L}es + {P}lans-sur-{B}ex, 1978)}, + SERIES = {Lecture Notes in Math.}, + VOLUME = {725}, + PAGES = {237--253}, + PUBLISHER = {Springer, Berlin}, + YEAR = {1979}, + ISBN = {3-540-09512-8}, + MRCLASS = {46L10 (46M20)}, + MRNUMBER = {548118}, +MRREVIEWER = {Alain\ Connes}, +} + +@article {Jones80finite, + AUTHOR = {Jones, Vaughan F. R.}, + TITLE = {Actions of finite groups on the hyperfinite type {II}$_1$ factor}, + JOURNAL = {Mem. Amer. Math. Soc.}, + FJOURNAL = {Memoirs of the American Mathematical Society}, + VOLUME = {28}, + YEAR = {1980}, + NUMBER = {237}, + PAGES = {v+70}, + ISSN = {0065-9266,1947-6221}, + MRCLASS = {46L55 (20C99 46L40)}, + MRNUMBER = {587749}, +MRREVIEWER = {Marie\ Choda}, + DOI = {10.1090/memo/0237}, + URL = {https://doi.org/10.1090/memo/0237}, +} + +@article {Jones80anti, + AUTHOR = {Jones, Vaughan F. R.}, + TITLE = {A {${\rm II}\sb{1}$} factor anti-isomorphic to itself but + without involutory antiautomorphisms}, + JOURNAL = {Math. Scand.}, + FJOURNAL = {Mathematica Scandinavica}, + VOLUME = {46}, + YEAR = {1980}, + NUMBER = {1}, + PAGES = {103--117}, + ISSN = {0025-5521,1903-1807}, + MRCLASS = {46L35}, + MRNUMBER = {585235}, +MRREVIEWER = {J.\ W.\ Bunce}, + DOI = {10.7146/math.scand.a-11855}, + URL = {https://doi.org/10.7146/math.scand.a-11855}, +} + +### KKKKKKKKKKKKKKKKKKKKKKKKK +# Kadison-Ringrose +@book {KadisonRingroseII, + AUTHOR = {Kadison, Richard V. and Ringrose, John R.}, + TITLE = {Fundamentals of the theory of operator algebras. {V}ol. {II}}, + SERIES = {Pure and Applied Mathematics}, + VOLUME = {100}, + NOTE = {Advanced theory}, + PUBLISHER = {Academic Press, Inc., Orlando, FL}, + YEAR = {1986}, + PAGES = {i--xiv and 399--1074}, + ISBN = {0-12-393302-1}, + MRCLASS = {46Lxx (46-01 46-02 47-01)}, + MRNUMBER = {859186}, +MRREVIEWER = {Robert\ S.\ Doran}, + DOI = {10.1016/S0079-8169(08)60611-X}, + URL = {https://doi.org/10.1016/S0079-8169(08)60611-X}, +} + + +# Kalantar-Kennedy +@article {KalantarKennedy17, + AUTHOR = {Kalantar, Mehrdad and Kennedy, Matthew}, + TITLE = {Boundaries of reduced {$C^*$}-algebras of discrete groups}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {727}, + YEAR = {2017}, + PAGES = {247--267}, + ISSN = {0075-4102,1435-5345}, + MRCLASS = {22D15 (20F67 22F05 46L10)}, + MRNUMBER = {3652252}, +MRREVIEWER = {Fernando\ Abadie}, + DOI = {10.1515/crelle-2014-0111}, + URL = {https://doi.org/10.1515/crelle-2014-0111}, +} + +# Kawahigashi +@article {Kawahigashi93, + AUTHOR = {Kawahigashi, Yasuyuki}, + TITLE = {Centrally trivial automorphisms and an analogue of {C}onnes's + {$\chi(M)$} for subfactors}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {71}, + YEAR = {1993}, + NUMBER = {1}, + PAGES = {93--118}, + ISSN = {0012-7094,1547-7398}, + MRCLASS = {46L40 (46L37)}, + MRNUMBER = {1230287}, +MRREVIEWER = {V.\ S.\ Sunder}, + DOI = {10.1215/S0012-7094-93-07105-0}, + URL = {https://doi.org/10.1215/S0012-7094-93-07105-0}, +} + +# Kawahigashi-Sutherland-Takesaki +@article {KawahigashiSutherlandTakesaki92, + AUTHOR = {Kawahigashi, Y. and Sutherland, C. E. and Takesaki, M.}, + TITLE = {The structure of the automorphism group of an injective factor + and the cocycle conjugacy of discrete abelian group actions}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {169}, + YEAR = {1992}, + NUMBER = {1-2}, + PAGES = {105--130}, + ISSN = {0001-5962,1871-2509}, + MRCLASS = {46L40}, + MRNUMBER = {1179014}, +MRREVIEWER = {Yoshikazu\ Katayama}, + DOI = {10.1007/BF02392758}, + URL = {https://doi.org/10.1007/BF02392758}, +} + + +# Kawamuro +@article {Kawamuro99, + AUTHOR = {Kawamuro, Keiko}, + TITLE = {Central sequence subfactors and double commutant properties}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {10}, + YEAR = {1999}, + NUMBER = {1}, + PAGES = {53--77}, + ISSN = {0129-167X}, + MRCLASS = {46L37}, + MRNUMBER = {1678538}, +MRREVIEWER = {Carl Winsl{\o}w}, + DOI = {10.1142/S0129167X99000033}, + URL = {https://doi.org/10.1142/S0129167X99000033}, +} + +# Kennedy +@article {Kennedy20, + AUTHOR = {Kennedy, Matthew}, + TITLE = {An intrinsic characterization of {$C^*$}-simplicity}, + JOURNAL = {Ann. Sci. \'Ec. Norm. Sup\'er. (4)}, + FJOURNAL = {Annales Scientifiques de l'\'Ecole Normale Sup\'erieure. + Quatri\`eme S\'erie}, + VOLUME = {53}, + YEAR = {2020}, + NUMBER = {5}, + PAGES = {1105--1119}, + ISSN = {0012-9593,1873-2151}, + MRCLASS = {20E07 (20C07 37A55 43A07 46L35)}, + MRNUMBER = {4174855}, +MRREVIEWER = {Sergey\ V.\ Ludkowski}, + DOI = {10.24033/asens.2441}, + URL = {https://doi.org/10.24033/asens.2441}, +} + + +# Kennedy-Schafhauser +@article {KennedySchafhauser19, + AUTHOR = {Kennedy, Matthew and Schafhauser, Christopher}, + TITLE = {Noncommutative boundaries and the ideal structure of reduced + crossed products}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {168}, + YEAR = {2019}, + NUMBER = {17}, + PAGES = {3215--3260}, + ISSN = {0012-7094,1547-7398}, + MRCLASS = {46L35 (43A65 47L65)}, + MRNUMBER = {4030364}, + DOI = {10.1215/00127094-2019-0032}, + URL = {https://doi.org/10.1215/00127094-2019-0032}, +} + +# Kennedy-Shamovich +@article {KennedyShamovich22, + AUTHOR = {Kennedy, Matthew and Shamovich, Eli}, + TITLE = {Noncommutative {C}hoquet simplices}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {382}, + YEAR = {2022}, + NUMBER = {3-4}, + PAGES = {1591--1629}, + ISSN = {0025-5831,1432-1807}, + MRCLASS = {46A55 (46L07 46L52 47A20 47L25)}, + MRNUMBER = {4403230}, +MRREVIEWER = {Andrey\ I.\ Zahariev}, + DOI = {10.1007/s00208-021-02261-z}, + URL = {https://doi.org/10.1007/s00208-021-02261-z}, +} + +# Kennedy-Ursu +@article{KennedyUrsu24, + title={Intermediate subalgebras for reduced crossed products of discrete groups}, + author={Kennedy, Matthew and Ursu, Dan}, + journal={arXiv preprint arXiv:2406.01546}, + year={2024} +} + +# Kerr +@article {Kerr20, + AUTHOR = {Kerr, David}, + TITLE = {Dimension, comparison, and almost finiteness}, + JOURNAL = {J. Eur. Math. Soc. (JEMS)}, + FJOURNAL = {Journal of the European Mathematical Society (JEMS)}, + VOLUME = {22}, + YEAR = {2020}, + NUMBER = {11}, + PAGES = {3697--3745}, + ISSN = {1435-9855,1435-9863}, + MRCLASS = {37A55 (37A20 46L35 46L55)}, + MRNUMBER = {4167017}, +MRREVIEWER = {Bruno\ Brogni\ Uggioni}, + DOI = {10.4171/jems/995}, + URL = {https://doi.org/10.4171/jems/995}, +} + +# Kervaire +@Inbook{Kervaire70, + author="Kervaire, Michel A.", + title="Multiplicateurs de Schur et K-th{\'e}orie", + bookTitle="Essays on Topology and Related Topics: Memoires d{\'e}di{\'e}s {\`a} Georges de Rham", + year="1970", + publisher="Springer Berlin Heidelberg", + address="Berlin, Heidelberg", + pages="212--225", + abstract="L'objet de cet article est de montrer que le groupe K2($\Lambda$) introduit par J. Milnor [3] s'interpr{\`e}te comme le multiplicateur de Schur (cf. [4]) du sous-groupe E($\Lambda$) de GL($\Lambda$) engendr{\'e} par les matrices {\'e}l{\'e}mentaires.", +isbn="978-3-642-49197-9", + doi="10.1007/978-3-642-49197-9_19", + url="https://doi.org/10.1007/978-3-642-49197-9_19" +} + + +# Kirchberg +@inproceedings {Kirchberg95, + AUTHOR = {Kirchberg, Eberhard}, + TITLE = {Exact {${\rm C}^*$}-algebras, tensor products, and the + classification of purely infinite algebras}, + BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of + {M}athematicians, {V}ol. 1, 2 ({Z}\"{u}rich, 1994)}, + PAGES = {943--954}, + PUBLISHER = {Birkh\"{a}user, Basel}, + YEAR = {1995}, + ISBN = {3-7643-5153-5}, + MRCLASS = {46L05 (46L35 46M05 46M15)}, + MRNUMBER = {1403994}, +MRREVIEWER = {Robert\ S.\ Doran}, +} + +@incollection {Kirchberg06, + AUTHOR = {Kirchberg, Eberhard}, + TITLE = {Central sequences in {C}*-algebras and strongly purely + infinite algebras}, + BOOKTITLE = {Operator {A}lgebras: {T}he {A}bel {S}ymposium 2004}, + SERIES = {Abel Symp.}, + VOLUME = {1}, + PAGES = {175--231}, + PUBLISHER = {Springer, Berlin}, + YEAR = {2006}, + MRCLASS = {46L05}, + MRNUMBER = {2265050}, +MRREVIEWER = {Wilhelm Winter}, + DOI = {10.1007/978-3-540-34197-0\_10}, + URL = {https://doi.org/10.1007/978-3-540-34197-0_10}, +} + +# Kirchberg-Phillips +@article {KirchbergPhillips00, + AUTHOR = {Kirchberg, Eberhard and Phillips, N. Christopher}, + TITLE = {Embedding of exact {C}*-algebras in the {C}untz algebra $\mathcal{O}_2$}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {525}, + YEAR = {2000}, + PAGES = {17--53}, + ISSN = {0075-4102}, + MRCLASS = {46L05 (19K56 46L35 46L80)}, + MRNUMBER = {1780426}, +MRREVIEWER = {Mikael R{\o}rdam}, + DOI = {10.1515/crll.2000.065}, + URL = {https://doi.org/10.1515/crll.2000.065}, +} + +# Kirchberg-Wassermann +@article {KirchbergWassermann99a, + AUTHOR = {Kirchberg, Eberhard and Wassermann, Simon}, + TITLE = {Exact groups and continuous bundles of {C}*-algebras}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {315}, + YEAR = {1999}, + NUMBER = {2}, + PAGES = {169--203}, + ISSN = {0025-5831,1432-1807}, + MRCLASS = {46L05 (22D05 46M20)}, + MRNUMBER = {1721796}, +MRREVIEWER = {Michael\ Frank}, + DOI = {10.1007/s002080050364}, + URL = {https://doi.org/10.1007/s002080050364}, +} + +@article {KirchbergWassermann99b, + AUTHOR = {Kirchberg, Eberhard and Wassermann, Simon}, + TITLE = {Permanence properties of {C}*-exact groups}, + JOURNAL = {Doc. Math.}, + FJOURNAL = {Documenta Mathematica}, + VOLUME = {4}, + YEAR = {1999}, + PAGES = {513--558}, + ISSN = {1431-0635,1431-0643}, + MRCLASS = {46L05 (22D05 46L80)}, + MRNUMBER = {1725812}, +MRREVIEWER = {Erik\ B\'{e}dos}, +} + + +# Kishimoto +@article {Kishimoto00, + AUTHOR = {Kishimoto, A.}, + TITLE = {{R}ohlin property for shift automorphisms}, + JOURNAL = {Rev. Math. Phys.}, + FJOURNAL = {Reviews in Mathematical Physics. A Journal for Both Review and + Original Research Papers in the Field of Mathematical Physics}, + VOLUME = {12}, + YEAR = {2000}, + NUMBER = {7}, + PAGES = {965--980}, + ISSN = {0129-055X}, + MRCLASS = {46L40 (46L05 46L55)}, + MRNUMBER = {1782691}, +MRREVIEWER = {Claire Anantharaman-Delaroche}, + DOI = {10.1142/S0129055X00000368}, + URL = {https://doi.org/10.1142/S0129055X00000368}, +} + +# Kuiper +@article {Kuiper65, + AUTHOR = {Kuiper, Nicolaas H.}, + TITLE = {The homotopy type of the unitary group of {H}ilbert space}, + JOURNAL = {Topology}, + FJOURNAL = {Topology. An International Journal of Mathematics}, + VOLUME = {3}, + YEAR = {1965}, + PAGES = {19--30}, + ISSN = {0040-9383}, + MRCLASS = {55.40}, + MRNUMBER = {179792}, +MRREVIEWER = {R.\ S.\ Palais}, + DOI = {10.1016/0040-9383(65)90067-4}, + URL = {https://doi.org/10.1016/0040-9383(65)90067-4}, +} + +# Kumjian +@article {Kumjian88, + AUTHOR = {Kumjian, Alexander}, + TITLE = {An involutive automorphism of the {B}unce-{D}eddens algebra}, + JOURNAL = {C. R. Math. Rep. Acad. Sci. Canada}, + FJOURNAL = {La Soci\'{e}t\'{e} Royale du Canada. L'Academie des Sciences. Comptes + Rendus Math\'{e}matiques. (Mathematical Reports)}, + VOLUME = {10}, + YEAR = {1988}, + NUMBER = {5}, + PAGES = {217--218}, + ISSN = {0706-1994}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {962104}, +MRREVIEWER = {J. W. Bunce}, +} + +### LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL +# Lee-Osaka +@article{LeeOsaka23, + title={On permanence of regularity properties}, + author={Lee, Hyun Ho and Osaka, Hiroyuki}, + journal={Journal of Topology and Analysis}, + year={2023}, + publisher={World Scientific} +} + +# Luck-Rordam +@article {LuckRordam93, + AUTHOR = {L\"{u}ck, Wolfgang and R{\o}rdam, Mikael}, + TITLE = {Algebraic {K}-theory of von {N}eumann algebras}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {7}, + YEAR = {1993}, + NUMBER = {6}, + PAGES = {517--536}, + ISSN = {0920-3036}, + MRCLASS = {46L80 (19B28 19J10 19K56 19K99)}, + MRNUMBER = {1268591}, +MRREVIEWER = {Jonathan M. Rosenberg}, + DOI = {10.1007/BF00961216}, + URL = {https://doi.org/10.1007/BF00961216}, +} + +### MMMMMMMMMMMMMMMMMMMMMMM +# Ma +@article {Ma21, + AUTHOR = {Ma, Xin}, + TITLE = {A generalized type semigroup and dynamical comparison}, + JOURNAL = {Ergodic Theory Dynam. Systems}, + FJOURNAL = {Ergodic Theory and Dynamical Systems}, + VOLUME = {41}, + YEAR = {2021}, + NUMBER = {7}, + PAGES = {2148--2165}, + ISSN = {0143-3857,1469-4417}, + MRCLASS = {37B05 (46L55)}, + MRNUMBER = {4266367}, +MRREVIEWER = {Geoffrey\ Price}, + DOI = {10.1017/etds.2020.28}, + URL = {https://doi.org/10.1017/etds.2020.28}, +} + +# Maclane +@book {Maclane12, + AUTHOR = {MacLane, Saunders}, + TITLE = {Homology}, + SERIES = {Die Grundlehren der mathematischen Wissenschaften, Band 114}, + EDITION = {first}, + PUBLISHER = {Springer-Verlag, Berlin-New York}, + YEAR = {1967}, + PAGES = {x+422}, + MRCLASS = {18-02}, + MRNUMBER = {0349792}, +} + +#Manor +@article {Manor21, + AUTHOR = {Manor, Nicholas}, + TITLE = {Exactness versus {C}*-exactness for certain + non-discrete groups}, + JOURNAL = {Integral Equations Operator Theory}, + FJOURNAL = {Integral Equations and Operator Theory}, + VOLUME = {93}, + YEAR = {2021}, + NUMBER = {3}, + PAGES = {Paper No. 20, 13}, + ISSN = {0378-620X,1420-8989}, + MRCLASS = {22D25 (46L06)}, + MRNUMBER = {4249041}, +MRREVIEWER = {Vladimir\ Manuilov}, + DOI = {10.1007/s00020-021-02634-8}, + URL = {https://doi.org/10.1007/s00020-021-02634-8}, +} + +# Marcoux +@article {Marcoux06, + AUTHOR = {Marcoux, Laurent W.}, + TITLE = {Sums of small number of commutators}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {56}, + YEAR = {2006}, + NUMBER = {1}, + PAGES = {111--142}, + ISSN = {0379-4024,1841-7744}, + MRCLASS = {46L05 (46L30 47B47)}, + MRNUMBER = {2261614}, +MRREVIEWER = {Steffen\ Roch}, +} + +# Marcoux-Popov +@article {MarcouxPopov16, + AUTHOR = {Marcoux, Laurent W. and Popov, Alexey I.}, + TITLE = {Abelian, amenable operator algebras are similar to + {C}*-algebras}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {165}, + YEAR = {2016}, + NUMBER = {12}, + PAGES = {2391--2406}, + ISSN = {0012-7094,1547-7398}, + MRCLASS = {46J05 (47L10 47L30)}, + MRNUMBER = {3544284}, +MRREVIEWER = {Damon\ Martin\ Hay}, + DOI = {10.1215/00127094-3619791}, + URL = {https://doi.org/10.1215/00127094-3619791}, +} + + +# Matui-Sato +@article {MatuiSato12, + AUTHOR = {Matui, Hiroki and Sato, Yasuhiko}, + TITLE = {Strict comparison and $\mathcal{Z}$-absorption of nuclear {C}*-algebras}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {209}, + YEAR = {2012}, + NUMBER = {1}, + PAGES = {179--196}, + ISSN = {0001-5962}, + MRCLASS = {46L05}, + MRNUMBER = {2979512}, +MRREVIEWER = {Caleb Eckhardt}, + DOI = {10.1007/s11511-012-0084-4}, + URL = {https://doi.org/10.1007/s11511-012-0084-4}, +} + +#McDuff +@article {McDuff69, + AUTHOR = {McDuff, Dusa}, + TITLE = {Uncountably many {II}$_1$ factors}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {90}, + YEAR = {1969}, + PAGES = {372--377}, + ISSN = {0003-486X}, + MRCLASS = {46.65}, + MRNUMBER = {259625}, +MRREVIEWER = {Z. Takeda}, + DOI = {10.2307/1970730}, + URL = {https://doi.org/10.2307/1970730}, +} + +@article {McDuff70, + AUTHOR = {McDuff, Dusa}, + TITLE = {Central sequences and the hyperfinite factor}, + JOURNAL = {Proc. London Math. Soc. (3)}, + FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, + VOLUME = {21}, + YEAR = {1970}, + PAGES = {443--461}, + ISSN = {0024-6115}, + MRCLASS = {46.65}, + MRNUMBER = {281018}, +MRREVIEWER = {M. Takesaki}, + DOI = {10.1112/plms/s3-21.3.443}, + URL = {https://doi.org/10.1112/plms/s3-21.3.443}, +} + +# Munkres +@book {Munkres, + AUTHOR = {Munkres, James R.}, + TITLE = {Topology}, + NOTE = {Second edition of [ MR0464128]}, + PUBLISHER = {Prentice Hall, Inc., Upper Saddle River, NJ}, + YEAR = {2000}, + PAGES = {xvi+537}, + ISBN = {0-13-181629-2}, + MRCLASS = {54-01}, + MRNUMBER = {3728284}, +} + +# Murphy +@book {Murphybook, + AUTHOR = {Murphy, Gerard J.}, + TITLE = {{C}*-algebras and operator theory}, + PUBLISHER = {Academic Press, Inc., Boston, MA}, + YEAR = {1990}, + PAGES = {x+286}, + ISBN = {0-12-511360-9}, + MRCLASS = {46Lxx (46-01)}, + MRNUMBER = {1074574}, +MRREVIEWER = {E. Gerlach}, +} + +# Murray-von Neumann +@article {MvNI, + AUTHOR = {Murray, Francis J. and von Neumann, John}, + TITLE = {On rings of operators}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {37}, + YEAR = {1936}, + NUMBER = {1}, + PAGES = {116--229}, + ISSN = {0003-486X,1939-8980}, + MRCLASS = {99-04}, + MRNUMBER = {1503275}, + DOI = {10.2307/1968693}, + URL = {https://doi.org/10.2307/1968693}, +} + +@article {MvNIV, + AUTHOR = {Murray, Francis J. and von Neumann, John}, + TITLE = {On rings of operators. {IV}}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {44}, + YEAR = {1943}, + PAGES = {716--808}, + ISSN = {0003-486X}, + MRCLASS = {46.0X}, + MRNUMBER = {9096}, +MRREVIEWER = {E. R. Lorch}, + DOI = {10.2307/1969107}, + URL = {https://doi.org/10.2307/1969107}, +} + +# Mygind +@article{Mygind01, + title={Classification of certain simple {C}*-algebras with torsion in {K}1}, + author={Mygind, Jesper}, + journal={Canadian Journal of Mathematics}, + volume={53}, + number={6}, + pages={1223--1308}, + year={2001}, + publisher={Cambridge University Press} +} +@article {MR1863849, + AUTHOR = {Mygind, Jesper}, + TITLE = {Classification of certain simple {C}*-algebras with torsion in {$K_1$}}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {53}, + YEAR = {2001}, + NUMBER = {6}, + PAGES = {1223--1308}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (19K14 46L80)}, + MRNUMBER = {1863849}, +MRREVIEWER = {Xiao Chun Fang}, + DOI = {10.4153/CJM-2001-046-2}, + URL = {https://doi.org/10.4153/CJM-2001-046-2}, +} + +### NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +# Nakamura +@article {Nakamura54, + AUTHOR = {Nakamura, Masahiro}, + TITLE = {On the direct product of finite factors}, + JOURNAL = {Tohoku Math. J. (2)}, + FJOURNAL = {The Tohoku Mathematical Journal. Second Series}, + VOLUME = {6}, + YEAR = {1954}, + PAGES = {205--207}, + ISSN = {0040-8735,2186-585X}, + MRCLASS = {46.0X}, + MRNUMBER = {70065}, +MRREVIEWER = {I.\ E.\ Segal}, + DOI = {10.2748/tmj/1178245180}, + URL = {https://doi.org/10.2748/tmj/1178245180}, +} + +# Ng +@article {Ng06, + AUTHOR = {Ng, Ping W.}, + TITLE = {Amenability of the sequence of unitary groups associated with + a {$C^*$}-algebra}, + JOURNAL = {Indiana Univ. Math. J.}, + FJOURNAL = {Indiana University Mathematics Journal}, + VOLUME = {55}, + YEAR = {2006}, + NUMBER = {4}, + PAGES = {1389--1400}, + ISSN = {0022-2518,1943-5258}, + MRCLASS = {46L05 (43A07)}, + MRNUMBER = {2269417}, +MRREVIEWER = {A.\ G.\ Myasnikov}, + DOI = {10.1512/iumj.2006.55.2791}, + URL = {https://doi.org/10.1512/iumj.2006.55.2791}, +} + +@article {Ng14, + AUTHOR = {Ng, Ping W.}, + TITLE = {The kernel of the determinant map on certain simple {C}*-algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {71}, + YEAR = {2014}, + NUMBER = {2}, + PAGES = {341--379}, + ISSN = {0379-4024}, + MRCLASS = {46L05 (46L80 47B47 47C15)}, + MRNUMBER = {3214642}, +MRREVIEWER = {Takahiro Sudo}, + DOI = {10.7900/jot.2012apr01.1953}, + URL = {https://doi.org/10.7900/jot.2012apr01.1953}, +} + +# Ng-Robert +@article {NgRobert16, + AUTHOR = {Ng, Ping W. and Robert, Leonel}, + TITLE = {Sums of commutators in pure {C}*-algebras}, + JOURNAL = {M\"{u}nster J. Math.}, + FJOURNAL = {M\"{u}nszter Journal of Mathematics}, + VOLUME = {9}, + YEAR = {2016}, + NUMBER = {1}, + PAGES = {121--154}, + ISSN = {1867-5778,1867-5786}, + MRCLASS = {46L05 (46L35 46L57 47B47)}, + MRNUMBER = {3549546}, +MRREVIEWER = {Bojan\ P.\ Magajna}, + DOI = {10.17879/35209721075}, + URL = {https://doi.org/10.17879/35209721075}, +} + +@article {NgRobert17, + AUTHOR = {Ng, Ping W. and Robert, Leonel}, + TITLE = {The kernel of the determinant map on pure {C}*-algebras}, + JOURNAL = {Houston J. Math.}, + FJOURNAL = {Houston Journal of Mathematics}, + VOLUME = {43}, + YEAR = {2017}, + NUMBER = {1}, + PAGES = {139--168}, + ISSN = {0362-1588}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {3647937}, +MRREVIEWER = {Daniele Puglisi}, + DOI = {10.1007/s11139-016-9879-9}, + URL = {https://doi.org/10.1007/s11139-016-9879-9}, +} + +# Nielsen-Thomsen +@article {NielsenThomsen96, + AUTHOR = {Nielsen, Karen E. and Thomsen, Klaus}, + TITLE = {Limits of circle algebras}, + JOURNAL = {Exposition. Math.}, + FJOURNAL = {Expositiones Mathematicae. International Journal}, + VOLUME = {14}, + YEAR = {1996}, + NUMBER = {1}, + PAGES = {17--56}, + ISSN = {0723-0869}, + MRCLASS = {46L80 (19K14 46L05)}, + MRNUMBER = {1382013}, +MRREVIEWER = {Sze-Kai Tsui}, +} + +### OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO + +@book {Ocneanu85, + AUTHOR = {Ocneanu, Adrian}, + TITLE = {Actions of discrete amenable groups on von {N}eumann algebras}, + SERIES = {Lecture Notes in Mathematics}, + VOLUME = {1138}, + PUBLISHER = {Springer-Verlag, Berlin}, + YEAR = {1985}, + PAGES = {iv+115}, + ISBN = {3-540-15663-1}, + MRCLASS = {46L55 (46L35 46L40)}, + MRNUMBER = {807949}, +MRREVIEWER = {Hisashi\ Choda}, + DOI = {10.1007/BFb0098579}, + URL = {https://doi.org/10.1007/BFb0098579}, +} + +@article {OsakaTeruya18, + AUTHOR = {Osaka, Hiroyuki and Teruya, Tamotsu}, + TITLE = {The {J}iang--{S}u absorption for inclusions of unital {C}*-algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {70}, + YEAR = {2018}, + NUMBER = {2}, + PAGES = {400--425}, + ISSN = {0008-414X}, + MRCLASS = {46L55 (46L35)}, + MRNUMBER = {3759005}, +MRREVIEWER = {Daniele Puglisi}, + DOI = {10.4153/CJM-2017-033-7}, + URL = {https://doi.org/10.4153/CJM-2017-033-7}, +} + +# Ozawa +@article {Ozawa13, + AUTHOR = {Ozawa, Narutaka}, + TITLE = {{D}ixmier approximation and symmetric amenability for {C}*-algebras}, + JOURNAL = {J. Math. Sci. Univ. Tokyo}, + FJOURNAL = {The University of Tokyo. Journal of Mathematical Sciences}, + VOLUME = {20}, + YEAR = {2013}, + NUMBER = {3}, + PAGES = {349--374}, + ISSN = {1340-5705}, + MRCLASS = {46L05 (46L10)}, + MRNUMBER = {3156986}, +MRREVIEWER = {William Paschke}, +} + +@article{Ozawa23, + title={Amenability for unitary groups of simple monotracial {C}*-algebras}, + author={Ozawa, Narutaka}, + journal={arXiv:2307.08267}, + year={2023} +} + +### PPPPPPPPPPPPPPPPPPPPPPP +# Paterson +@article{Paterson83, + AUTHOR = {Paterson, Alan L. T.}, + TITLE = {Harmonic analysis on unitary groups}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {53}, + YEAR = {1983}, + NUMBER = {3}, + PAGES = {203--223}, + ISSN = {0022-1236}, + MRCLASS = {22D10 (22D25 43A65 46L05)}, + MRNUMBER = {724026}, +MRREVIEWER = {J. Gil de Lamadrid}, + DOI = {10.1016/0022-1236(83)90031-9}, + URL = {https://doi.org/10.1016/0022-1236(83)90031-9}, +} + +@article {Paterson92, + AUTHOR = {Paterson, Alan L. T.}, + TITLE = {Nuclear {C}*-algebras have amenable unitary groups}, + JOURNAL = {Proc. Amer. Math. Soc.}, + FJOURNAL = {Proceedings of the American Mathematical Society}, + VOLUME = {114}, + YEAR = {1992}, + NUMBER = {3}, + PAGES = {719--721}, + ISSN = {0002-9939}, + MRCLASS = {46L05}, + MRNUMBER = {1076577}, +MRREVIEWER = {C. J. K. Batty}, + DOI = {10.2307/2159395}, + URL = {https://doi.org/10.2307/2159395}, +} + +# Paulsen +@book{Paulsenbook, + AUTHOR = {Paulsen, Vern}, + TITLE = {Completely bounded maps and operator algebras}, + SERIES = {Cambridge Studies in Advanced Mathematics}, + VOLUME = {78}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {2002}, + PAGES = {xii+300}, + ISBN = {0-521-81669-6}, + MRCLASS = {46L07 (47A20 47L30)}, + MRNUMBER = {1976867}, +MRREVIEWER = {Christian Le Merdy}, +} + +# Phillips +@article {Phillips92, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {The rectifiable metric on the space of projections in a {C}*-algebra}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {3}, + YEAR = {1992}, + NUMBER = {5}, + PAGES = {679--698}, + ISSN = {0129-167X}, + MRCLASS = {46L05}, + MRNUMBER = {1189681}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1142/S0129167X92000333}, + URL = {https://doi.org/10.1142/S0129167X92000333}, +} + +@article {Phillips95, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {Exponential length and traces}, + JOURNAL = {Proc. Roy. Soc. Edinburgh Sect. A}, + FJOURNAL = {Proceedings of the Royal Society of Edinburgh. Section A. + Mathematics}, + VOLUME = {125}, + YEAR = {1995}, + NUMBER = {1}, + PAGES = {13--29}, + ISSN = {0308-2105}, + MRCLASS = {46L05}, + MRNUMBER = {1318621}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1017/S0308210500030730}, + URL = {https://doi.org/10.1017/S0308210500030730}, +} + +@article {Phillips00, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {A classification theorem for nuclear purely infinite simple {C}*-algebras}, + JOURNAL = {Doc. Math.}, + FJOURNAL = {Documenta Mathematica}, + VOLUME = {5}, + YEAR = {2000}, + PAGES = {49--114}, + ISSN = {1431-0635}, + MRCLASS = {46L05 (19K56 46L35 46L80)}, + MRNUMBER = {1745197}, +MRREVIEWER = {Mikael R{\o}rdam}, +} + +@article {Phillips01, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {Continuous-trace {C}*-algebras not isomorphic to their opposite algebras}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {12}, + YEAR = {2001}, + NUMBER = {3}, + PAGES = {263--275}, + ISSN = {0129-167X}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {1841515}, +MRREVIEWER = {Takahiro Sudo}, + DOI = {10.1142/S0129167X01000642}, + URL = {https://doi.org/10.1142/S0129167X01000642}, +} + +@article{Phillips04, + title={A simple separable {C}*-algebra not isomorphic to its opposite algebra}, + author={Phillips, N. Christopher}, + journal={Proceedings of the American Mathematical Society}, + volume={132}, + number={10}, + pages={2997--3005}, + year={2004} +} +@article {Phillips04, + AUTHOR = {Phillips, N. Christopher}, + TITLE = {A simple separable {C}*-algebra not isomorphic to its opposite algebra}, + JOURNAL = {Proc. Amer. Math. Soc.}, + FJOURNAL = {Proceedings of the American Mathematical Society}, + VOLUME = {132}, + YEAR = {2004}, + NUMBER = {10}, + PAGES = {2997--3005}, + ISSN = {0002-9939}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {2063121}, +MRREVIEWER = {Hiroyuki Osaka}, + DOI = {10.1090/S0002-9939-04-07330-7}, + URL = {https://doi.org/10.1090/S0002-9939-04-07330-7}, +} + +@article{Phillips12, + title={The tracial {R}okhlin property is generic}, + author={Phillips, N. Christopher}, + journal={arXiv:1209.3859}, + year={2012} +} + +# Phillips-Viola +@article {PhillipsViola13, + AUTHOR = {Phillips, N. Christopher and Viola, Maria Grazia}, + TITLE = {A simple separable exact {C}*-algebra not anti-isomorphic to itself}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {355}, + YEAR = {2013}, + NUMBER = {2}, + PAGES = {783--799}, + ISSN = {0025-5831}, + MRCLASS = {46L35 (46L09 46L37 46L40)}, + MRNUMBER = {3010147}, +MRREVIEWER = {Snigdhayan Mahanta}, + DOI = {10.1007/s00208-011-0755-z}, + URL = {https://doi.org/10.1007/s00208-011-0755-z}, +} + +# Pimsner-Voiculescu +@article {PimsnerVoiculescu80, + AUTHOR = {Pimsner, Mihai and Voiculescu, Dan}, + TITLE = {Imbedding the irrational rotation {C}*-algebra into {AF}-algebra}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {4}, + YEAR = {1980}, + NUMBER = {2}, + PAGES = {201--210}, + ISSN = {0379-4024}, + MRCLASS = {46L35 (16A54)}, + MRNUMBER = {595412}, +MRREVIEWER = {David Handelman}, +} + +# Pitts +@article {Pitts17, + AUTHOR = {Pitts, David R.}, + TITLE = {Structure for regular inclusions. {I}}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {78}, + YEAR = {2017}, + NUMBER = {2}, + PAGES = {357--416}, + ISSN = {0379-4024,1841-7744}, + MRCLASS = {46L05 (47L30)}, + MRNUMBER = {3725511}, +MRREVIEWER = {Michael\ S.\ Anoussis}, + DOI = {10.7900/jot}, + URL = {https://doi.org/10.7900/jot}, +} + +@article {Pitts21StructureII, + AUTHOR = {Pitts, David R.}, + TITLE = {Structure for regular inclusions. {II}: {C}artan envelopes, + pseudo-expectations and twists}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {281}, + YEAR = {2021}, + NUMBER = {1}, + PAGES = {Paper No. 108993, 66}, + ISSN = {0022-1236,1096-0783}, + MRCLASS = {46L05 (22A22 46L07)}, + MRNUMBER = {4234857}, +MRREVIEWER = {Preeti\ Luthra}, + DOI = {10.1016/j.jfa.2021.108993}, + URL = {https://doi.org/10.1016/j.jfa.2021.108993}, +} + +@article{Pitts21Normalizers, + title={Normalizers and approximate units for inclusions of C*-algebras}, + author={Pitts, David R}, + journal={arXiv:2109.00856}, + year={2021} +} + +# Pitts-Smith-Zarikian +@article{PittsSmithZarikian23, + title={Norming in Discrete Crossed Products}, + author={Pitts, David R and Smith, Roger R and Zarikian, Vrej}, + journal={arXiv:2307.15571}, + year={2023} +} + +# Pitts-Zarikian +@article {PittsZarikian15, + AUTHOR = {Pitts, David R. and Zarikian, Vrej}, + TITLE = {Unique pseudo-expectations for {C}*-inclusions}, + JOURNAL = {Illinois J. Math.}, + FJOURNAL = {Illinois Journal of Mathematics}, + VOLUME = {59}, + YEAR = {2015}, + NUMBER = {2}, + PAGES = {449--483}, + ISSN = {0019-2082,1945-6581}, + MRCLASS = {46L05 (46L07 46L10 46M10)}, + MRNUMBER = {3499520}, +MRREVIEWER = {Bernard\ Russo}, + URL = {http://projecteuclid.org/euclid.ijm/1462450709}, +} + +# Popa +@article {Popa83, + AUTHOR = {Popa, Sorin}, + TITLE = {Orthogonal pairs of *-subalgebras in finite von {N}eumann algebras}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {9}, + YEAR = {1983}, + NUMBER = {2}, + PAGES = {253--268}, + ISSN = {0379-4024}, + MRCLASS = {46L10 (16A30 20C07)}, + MRNUMBER = {703810}, +MRREVIEWER = {G. A. Elliott}, +} + + +@article {Popa89, + AUTHOR = {Popa, Sorin}, + TITLE = {Sousfacteurs, actions des groupes et cohomologie}, + JOURNAL = {C. R. Acad. Sci. Paris S\'{e}r. I Math.}, + FJOURNAL = {Comptes Rendus de l'Acad\'{e}mie des Sciences. S\'{e}rie I. + Math\'{e}matique}, + VOLUME = {309}, + YEAR = {1989}, + NUMBER = {12}, + PAGES = {771--776}, + ISSN = {0764-4442}, + MRCLASS = {46L37 (22D25 22D35 46L55 46L80)}, + MRNUMBER = {1054961}, +MRREVIEWER = {Masatoshi Enomoto}, +} + +@article {Popa00, + AUTHOR = {Popa, Sorin}, + TITLE = {On the relative {D}ixmier property for inclusions of {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {171}, + YEAR = {2000}, + NUMBER = {1}, + PAGES = {139--154}, + ISSN = {0022-1236}, + MRCLASS = {46L37 (46L05)}, + MRNUMBER = {1742862}, +MRREVIEWER = {H. Halpern}, + DOI = {10.1006/jfan.1999.3536}, + URL = {https://doi.org/10.1006/jfan.1999.3536}, +} + +# Popa-Takesaki +@article {PopaTakesaki93, + AUTHOR = {Popa, Sorin and Takesaki, Masamichi}, + TITLE = {The topological structure of the unitary and automorphism + groups of a factor}, + JOURNAL = {Comm. Math. Phys.}, + FJOURNAL = {Communications in Mathematical Physics}, + VOLUME = {155}, + YEAR = {1993}, + NUMBER = {1}, + PAGES = {93--101}, + ISSN = {0010-3616,1432-0916}, + MRCLASS = {46L10 (46L40)}, + MRNUMBER = {1228527}, +MRREVIEWER = {Yasuyuki\ Kawahigashi}, + URL = {http://projecteuclid.org/euclid.cmp/1104253201}, +} + +### QQQQQQQQQQQQQQQQQQQQQQQQ +# Quillen +@incollection {Quillen73, + AUTHOR = {Quillen, Daniel}, + TITLE = {Higher algebraic {$K$}-theory. {I}}, + BOOKTITLE = {Algebraic {$K$}-theory, {I}: {H}igher {$K$}-theories ({P}roc. + {C}onf., {B}attelle {M}emorial {I}nst., {S}eattle, {W}ash., + 1972)}, + SERIES = {Lecture Notes in Math.}, + VOLUME = {Vol. 341}, + PAGES = {85--147}, + PUBLISHER = {Springer, Berlin-New York}, + YEAR = {1973}, + MRCLASS = {18F25}, + MRNUMBER = {338129}, +MRREVIEWER = {Stephen\ M.\ Gersten}, +} + +### RRRRRRRRRRRRRRRRRRRRRRRRRR +# Raeburn +@book {RaeburnBook, + AUTHOR = {Raeburn, Iain}, + TITLE = {Graph algebras}, + SERIES = {CBMS Regional Conference Series in Mathematics}, + VOLUME = {103}, + PUBLISHER = {Published for the Conference Board of the Mathematical + Sciences, Washington, DC; by the American Mathematical + Society, Providence, RI}, + YEAR = {2005}, + PAGES = {vi+113}, + ISBN = {0-8218-3660-9}, + MRCLASS = {46L05 (22D25)}, + MRNUMBER = {2135030}, +MRREVIEWER = {Mark Tomforde}, + DOI = {10.1090/cbms/103}, + URL = {https://doi.org/10.1090/cbms/103}, +} + +#Rieffel +@article {Rieffel83, + AUTHOR = {Rieffel, Marc A.}, + TITLE = {Dimension and stable rank in the {$K$}-theory of {C}*-algebras}, + JOURNAL = {Proc. London Math. Soc. (3)}, + FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, + VOLUME = {46}, + YEAR = {1983}, + NUMBER = {2}, + PAGES = {301--333}, + ISSN = {0024-6115,1460-244X}, + MRCLASS = {46L05 (46H20 46M20)}, + MRNUMBER = {693043}, +MRREVIEWER = {David\ Handelman}, + DOI = {10.1112/plms/s3-46.2.301}, + URL = {https://doi.org/10.1112/plms/s3-46.2.301}, +} + + +@article {Rieffel87, + AUTHOR = {Rieffel, Marc A.}, + TITLE = {The homotopy groups of the unitary groups of noncommutative tori}, + JOURNAL = {J. Operator Theory}, + FJOURNAL = {Journal of Operator Theory}, + VOLUME = {17}, + YEAR = {1987}, + NUMBER = {2}, + PAGES = {237--254}, + ISSN = {0379-4024}, + MRCLASS = {22D25 (46L55 46L80)}, + MRNUMBER = {887221}, +MRREVIEWER = {Gustavo Corach}, +} + +# Robert +@article {Robert12, + AUTHOR = {Robert, Leonel}, + TITLE = {Classification of inductive limits of 1-dimensional {NCCW} + complexes}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {231}, + YEAR = {2012}, + NUMBER = {5}, + PAGES = {2802--2836}, + ISSN = {0001-8708}, + MRCLASS = {46L35}, + MRNUMBER = {2970466}, +MRREVIEWER = {Changguo Wei}, + DOI = {10.1016/j.aim.2012.07.010}, + URL = {https://doi.org/10.1016/j.aim.2012.07.010}, +} + +@article {Robert19, + AUTHOR = {Robert, Leonel}, + TITLE = {Normal subgroups of invertibles and of unitaries in a {C}*-algebra}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {756}, + YEAR = {2019}, + PAGES = {285--319}, + ISSN = {0075-4102,1435-5345}, + MRCLASS = {46L05 (46L40)}, + MRNUMBER = {4026455}, +MRREVIEWER = {Jia-jie\ Hua}, + DOI = {10.1515/crelle-2017-0020}, + URL = {https://doi.org/10.1515/crelle-2017-0020}, +} + +#Robertson +@article {Robertson80, + AUTHOR = {Robertson, A. Guyan}, + TITLE = {Stable range in {C}*-algebras}, + JOURNAL = {Math. Proc. Cambridge Philos. Soc.}, + FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical + Society}, + VOLUME = {87}, + YEAR = {1980}, + NUMBER = {3}, + PAGES = {413--418}, + ISSN = {0305-0041,1469-8064}, + MRCLASS = {46L05}, + MRNUMBER = {556921}, + DOI = {10.1017/S030500410005684X}, + URL = {https://doi.org/10.1017/S030500410005684X}, +} + + +# Rordam +@article {Rordam91, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {On the structure of simple {C}*-algebras tensored with a {UHF}-algebra}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {100}, + YEAR = {1991}, + NUMBER = {1}, + PAGES = {1--17}, + ISSN = {0022-1236}, + MRCLASS = {46L05}, + MRNUMBER = {1124289}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.1016/0022-1236(91)90098-P}, + URL = {https://doi.org/10.1016/0022-1236(91)90098-P}, +} + + +@article {Rordam92, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {On the structure of simple {C}*-algebras tensored with a {UHF}-algebra. {II}}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {107}, + YEAR = {1992}, + NUMBER = {2}, + PAGES = {255--269}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L85)}, + MRNUMBER = {1172023}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.1016/0022-1236(92)90106-S}, + URL = {https://doi.org/10.1016/0022-1236(92)90106-S}, +} + +@article {Rordam93, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Classification of inductive limits of {C}untz algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {440}, + YEAR = {1993}, + PAGES = {175--200}, + ISSN = {0075-4102}, + MRCLASS = {46L05 (19K99 46L80)}, + MRNUMBER = {1225963}, +MRREVIEWER = {Shuang Zhang}, + DOI = {10.1515/crll.1993.440.175}, + URL = {https://doi.org/10.1515/crll.1993.440.175}, +} + +@article {Rordam95, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Classification of certain infinite simple {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {131}, + YEAR = {1995}, + NUMBER = {2}, + PAGES = {415--458}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (19K35 46L35 46L80)}, + MRNUMBER = {1345038}, +MRREVIEWER = {Judith A. Packer}, + DOI = {10.1006/jfan.1995.1095}, + URL = {https://doi.org/10.1006/jfan.1995.1095}, +} + +@incollection {RordamBook, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Classification of nuclear, simple {C}*-algebras}, + BOOKTITLE = {Classification of nuclear {C}*-algebras. {E}ntropy in + operator algebras}, + SERIES = {Encyclopaedia Math. Sci.}, + VOLUME = {126}, + PAGES = {1--145}, + PUBLISHER = {Springer, Berlin}, + YEAR = {2002}, + MRCLASS = {46L05 (19K56 46L35 46L80)}, + MRNUMBER = {1878882}, +MRREVIEWER = {Judith A. Packer}, + DOI = {10.1007/978-3-662-04825-2\_1}, + URL = {https://doi.org/10.1007/978-3-662-04825-2_1}, +} + +@book {RordamKBook, + AUTHOR = {R{\o}rdam, Mikael and Larsen, Flemming and Laustsen, Niels J.}, + TITLE = {An introduction to {K}-theory for {C}*-algebras}, + SERIES = {London Mathematical Society Student Texts}, + VOLUME = {49}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {2000}, + PAGES = {xii+242}, + ISBN = {0-521-78334-8; 0-521-78944-3}, + MRCLASS = {46-01 (19K35 46L80)}, + MRNUMBER = {1783408}, +MRREVIEWER = {\'{E}ric Leichtnam}, +} + +@article {Rordam03, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {A simple {C}*-algebra with a finite and an infinite projection}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {191}, + YEAR = {2003}, + NUMBER = {1}, + PAGES = {109--142}, + ISSN = {0001-5962}, + MRCLASS = {46L05}, + MRNUMBER = {2020420}, +MRREVIEWER = {Berndt Brenken}, + DOI = {10.1007/BF02392697}, + URL = {https://doi.org/10.1007/BF02392697}, +} + +@article {Rordam04, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {The stable and the real rank of $\mathcal{Z}$-absorbing {C}*-algebras}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {15}, + YEAR = {2004}, + NUMBER = {10}, + PAGES = {1065--1084}, + ISSN = {0129-167X}, + MRCLASS = {46L35 (19K14 46L05 46L06)}, + MRNUMBER = {2106263}, +MRREVIEWER = {Nadia S. Larsen}, + DOI = {10.1142/S0129167X04002661}, + URL = {https://doi.org/10.1142/S0129167X04002661}, +} + +@article {Rordam23, + AUTHOR = {R{\o}rdam, Mikael}, + TITLE = {Irreducible inclusions of simple {C}*-algebras}, + JOURNAL = {Enseign. Math.}, + FJOURNAL = {L'Enseignement Math\'{e}matique}, + VOLUME = {69}, + YEAR = {2023}, + NUMBER = {3-4}, + PAGES = {275--314}, + ISSN = {0013-8584,2309-4672}, + MRCLASS = {46L05 (46L35 46L55)}, + MRNUMBER = {4599249}, + DOI = {10.4171/lem/1051}, + URL = {https://doi.org/10.4171/lem/1051}, +} + +# Rordam-Winter +@article {RordamWinter10, + AUTHOR = {R{\o}rdam, Mikael and Winter, Wilhelm}, + TITLE = {The {J}iang-{S}u algebra revisited}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {642}, + YEAR = {2010}, + PAGES = {129--155}, + ISSN = {0075-4102}, + MRCLASS = {46L35 (46L05 46L85)}, + MRNUMBER = {2658184}, +MRREVIEWER = {Hiroyuki Osaka}, + DOI = {10.1515/CRELLE.2010.039}, + URL = {https://doi.org/10.1515/CRELLE.2010.039}, +} + +# Rotman +@book {RotmanATBook, + AUTHOR = {Rotman, Joseph J.}, + TITLE = {An introduction to algebraic topology}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {119}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1988}, + PAGES = {xiv+433}, + ISBN = {0-387-96678-1}, + MRCLASS = {55-01}, + MRNUMBER = {957919}, +MRREVIEWER = {P.\ J.\ Kahn}, + DOI = {10.1007/978-1-4612-4576-6}, + URL = {https://doi.org/10.1007/978-1-4612-4576-6}, +} + + +# Rosenberg +@article {Rosenberg89, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {Continuous-trace algebras from the bundle theoretic point of + view}, + JOURNAL = {J. Austral. Math. Soc. Ser. A}, + FJOURNAL = {Australian Mathematical Society. Journal. Series A. Pure + Mathematics and Statistics}, + VOLUME = {47}, + YEAR = {1989}, + NUMBER = {3}, + PAGES = {368--381}, + ISSN = {0263-6115}, + MRCLASS = {46L80 (19K99 46L10 46M20 55R10)}, + MRNUMBER = {1018964}, +MRREVIEWER = {Claude Schochet}, +} + +@incollection {Rosenberg04, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {Comparison between algebraic and topological {K}-theory for + {B}anach algebras and {C}*-algebras}, + BOOKTITLE = {Handbook of {$K$}-theory. {V}ol. 1, 2}, + PAGES = {843--874}, + PUBLISHER = {Springer, Berlin}, + YEAR = {2005}, + MRCLASS = {46L80 (19K99 46H99)}, + MRNUMBER = {2181834}, +MRREVIEWER = {Efton Park}, + DOI = {10.1007/978-3-540-27855-9\_16}, + URL = {https://doi.org/10.1007/978-3-540-27855-9_16}, +} + + +@article {Rosenberg97, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {The algebraic {$K$}-theory of operator algebras}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {12}, + YEAR = {1997}, + NUMBER = {1}, + PAGES = {75--99}, + ISSN = {0920-3036}, + MRCLASS = {19K99 (19D35 19D50 19L41 46L80)}, + MRNUMBER = {1466624}, +MRREVIEWER = {Hiroshi Takai}, + DOI = {10.1023/A:1007736420938}, + URL = {https://doi.org/10.1023/A:1007736420938}, +} + +@book {Rosenberg94, + AUTHOR = {Rosenberg, Jonathan}, + TITLE = {Algebraic {$K$}-theory and its applications}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {147}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1994}, + PAGES = {x+392}, + ISBN = {0-387-94248-3}, + MRCLASS = {19-01 (19-02)}, + MRNUMBER = {1282290}, +MRREVIEWER = {Dominique Arlettaz}, + DOI = {10.1007/978-1-4612-4314-4}, + URL = {https://doi.org/10.1007/978-1-4612-4314-4}, +} + +# Rosenberg-Schochet +@article {RosenbergSchochet87, + AUTHOR = {Rosenberg, Jonathan and Schochet, Claude}, + TITLE = {The {K}\"{u}nneth theorem and the universal coefficient theorem + for {K}asparov's generalized {$K$}-functor}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {55}, + YEAR = {1987}, + NUMBER = {2}, + PAGES = {431--474}, + ISSN = {0012-7094}, + MRCLASS = {46L80 (19K33 46M20 58G12)}, + MRNUMBER = {894590}, +MRREVIEWER = {Thierry Fack}, + DOI = {10.1215/S0012-7094-87-05524-4}, + URL = {https://doi.org/10.1215/S0012-7094-87-05524-4}, +} + +# Russo-Dye +@article {RussoDye66, + AUTHOR = {Russo, Bernard and Dye, Henry A.}, + TITLE = {A note on unitary operators in {$C\sp{\ast} $}-algebras}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {33}, + YEAR = {1966}, + PAGES = {413--416}, + ISSN = {0012-7094,1547-7398}, + MRCLASS = {46.65}, + MRNUMBER = {193530}, +MRREVIEWER = {J.\ Glimm}, + URL = {http://projecteuclid.org/euclid.dmj/1077376395}, +} + + +### SSSSSSSSSSSSSSSSSSS +# Sakai +@article {Sakai55, + AUTHOR = {Sakai, Sh\^{o}ichir\^{o}}, + TITLE = {On the group isomorphism of unitary groups in {AW}*-algebras}, + JOURNAL = {Tohoku Math. J. (2)}, + FJOURNAL = {The Tohoku Mathematical Journal. Second Series}, + VOLUME = {7}, + YEAR = {1955}, + PAGES = {87--95}, + ISSN = {0040-8735}, + MRCLASS = {46.2X}, + MRNUMBER = {73139}, +MRREVIEWER = {J. Feldman}, + DOI = {10.2748/tmj/1178245106}, + URL = {https://doi.org/10.2748/tmj/1178245106}, +} + +@article {Sakai70, + AUTHOR = {Sakai, Sh\^{o}ichir\^{o}}, + TITLE = {An uncountable number of {II}$_1$ and {II}$_{\infty}$ factors}, + JOURNAL = {J. Functional Analysis}, + VOLUME = {5}, + YEAR = {1970}, + PAGES = {236--246}, + MRCLASS = {46.65}, + MRNUMBER = {0259626}, +MRREVIEWER = {Z. Takeda}, + DOI = {10.1016/0022-1236(70)90028-5}, + URL = {https://doi.org/10.1016/0022-1236(70)90028-5}, +} + +@book{Sakaibook, + title={{C}*-algebras and {W}*-algebras}, + author={Sakai, Sh\^{o}ichir\^{o}}, + year={2012}, + publisher={Springer Science \& Business Media} +} + +# Sarkowicz +@article{Sarkowicz24inclusion, + title={Tensorially absorbing inclusions of {C}*-algebras}, + DOI={10.4153/S0008414X24000324}, + journal={Canadian Journal of Mathematics}, + author={Pawel Sarkowicz}, + year={2024}, + note={published online: doi:10.4153/S0008414X24000324}, + pages={1–32} +} + +@article{Sarkowicz24unitary, + title={Unitary groups, {K}-theory, and traces}, + volume={66}, + DOI={10.1017/S0017089523000447}, + number={2}, + journal={Glasgow Mathematical Journal}, + author={Sarkowicz, Pawel}, + year={2024}, + pages={229–251} +} + +@article{Sarkowicz24covering, + title={Universal covering groups of unitary groups of von {N}eumann algebras}, + author={Pawel Sarkowicz}, + year={2024}, + eprint={2408.13710}, + archivePrefix={arXiv}, + primaryClass={math.OA}, + url={https://arxiv.org/abs/2408.13710}, + journal={arXiv preprint arXiv:2408.13710}, +} + +# Sarkowicz-Tikuisis +@article{SarkowiczTikuisis23, + title={Polar decomposition in algebraic {K}-theory}, + author={Pawel Sarkowicz and Aaron Tikuisis}, + year={2023}, + NOTE={to appear in Journal of Operator Theory} +} + + + +# Sato +@article {Sato10, + AUTHOR = {Sato, Yasuhiko}, + TITLE = {The {R}ohlin property for automorphisms of the {J}iang-{S}u + algebra}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {259}, + YEAR = {2010}, + NUMBER = {2}, + PAGES = {453--476}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L35 46L40 46L80)}, + MRNUMBER = {2644109}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.1016/j.jfa.2010.04.006}, + URL = {https://doi.org/10.1016/j.jfa.2010.04.006}, +} + +# Sato-White-Winter +@article {SatoWhiteWinter15, + AUTHOR = {Sato, Yasuhiko and White, Stuart and Winter, Wilhelm}, + TITLE = {Nuclear dimension and {$\mathcal{Z}$}-stability}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {202}, + YEAR = {2015}, + NUMBER = {2}, + PAGES = {893--921}, + ISSN = {0020-9910}, + MRCLASS = {46L35 (45L05)}, + MRNUMBER = {3418247}, +MRREVIEWER = {Aaron Tikuisis}, + DOI = {10.1007/s00222-015-0580-1}, + URL = {https://doi.org/10.1007/s00222-015-0580-1}, +} + +# Schafhauser +@article {Schafhauser20, + AUTHOR = {Schafhauser, Christopher}, + TITLE = {Subalgebras of simple {AF}-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {192}, + YEAR = {2020}, + NUMBER = {2}, + PAGES = {309--352}, + ISSN = {0003-486X}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {4151079}, +MRREVIEWER = {Daniele Puglisi}, + DOI = {10.4007/annals.2020.192.2.1}, + URL = {https://doi.org/10.4007/annals.2020.192.2.1}, +} + +# Schemaitat +@article {Schemaitat22, + AUTHOR = {Schemaitat, Andr\'{e}}, + TITLE = {The {J}iang-{S}u algebra is strongly self-absorbing revisited}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {282}, + YEAR = {2022}, + NUMBER = {6}, + PAGES = {Paper No. 109347, 39}, + ISSN = {0022-1236}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {4360358}, +MRREVIEWER = {Prahlad Vaidyanathan}, + DOI = {10.1016/j.jfa.2021.109347}, + URL = {https://doi.org/10.1016/j.jfa.2021.109347}, +} + +# Schochet +@article {SchochetIV, + AUTHOR = {Schochet, Claude}, + TITLE = {Topological methods for {C}*-algebras. {IV}. {M}od {$p$} homology}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {114}, + YEAR = {1984}, + NUMBER = {2}, + PAGES = {447--468}, + ISSN = {0030-8730}, + MRCLASS = {46L80 (19K33 46M20 55N99)}, + MRNUMBER = {757511}, +MRREVIEWER = {Vern Paulsen}, + URL = {http://projecteuclid.org/euclid.pjm/1102708718}, +} + +# Schröder +@article{Schroder84, + author = {Schr{\"o}der, Herbert}, + journal = {Mathematische Annalen}, + keywords = {finite continuous W*-algebra; group of regular elements; fibration; exact homotopy + sequence; periodicity theorem}, + pages = {271-278}, + title = {On the Homotopy Type of the Regular Group of a {W}*-Algebra.}, + url = {http://eudml.org/doc/163893}, + volume = {267}, + year = {1984}, +} + +@article{Schroder84real, + title={On the homotopy type of the regular group of a real {W}*-algebra}, + author={Schr{\"o}der, Herbert}, + journal={Integral Equations and Operator Theory}, + year={1984}, + volume={9}, + pages={694-705}, + url={https://api.semanticscholar.org/CorpusID:115847290} +} + +# Schur +@article {Schur04, + AUTHOR = {Schur, Issai}, + TITLE = {\"{U}ber die {D}arstellung der endlichen {G}ruppen durch + gebrochen lineare {S}ubstitutionen}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {127}, + YEAR = {1904}, + PAGES = {20--50}, + ISSN = {0075-4102,1435-5345}, + MRCLASS = {99-04}, + MRNUMBER = {1580631}, + DOI = {10.1515/crll.1904.127.20}, + URL = {https://doi.org/10.1515/crll.1904.127.20}, +} + +# Serre +@book {Serre77, + AUTHOR = {Serre, Jean-Pierre}, + TITLE = {Linear representations of finite groups}, + SERIES = {Graduate Texts in Mathematics, Vol. 42}, + NOTE = {Translated from the second French edition by Leonard L. Scott}, + PUBLISHER = {Springer-Verlag, New York-Heidelberg}, + YEAR = {1977}, + PAGES = {x+170}, + ISBN = {0-387-90190-6}, + MRCLASS = {20CXX}, + MRNUMBER = {0450380}, +MRREVIEWER = {W. Feit}, +} + +# Shoda +@article {Shoda37, + AUTHOR = {Shoda, Kenjiro}, + TITLE = {Einige {S}\"{a}tze \"{u}ber {M}atrizen}, + JOURNAL = {Jpn. J. Math.}, + FJOURNAL = {Japanese Journal of Mathematics}, + VOLUME = {13}, + YEAR = {1937}, + NUMBER = {3}, + PAGES = {361--365}, + ISSN = {0075-3432}, + MRCLASS = {15A24}, + MRNUMBER = {3223061}, + DOI = {10.4099/jjm1924.13.0\_361}, + URL = {https://doi.org/10.4099/jjm1924.13.0_361}, +} + +# Silverman +@book{Silverman14, + title={A friendly introduction to number theory}, + author={Silverman, Joseph H.}, + year={2014}, + publisher={Pearson} +} + +# Sinclair Smith +@book {SinclairSmithBook, + AUTHOR = {Sinclair, Allan M. and Smith, Roger R.}, + TITLE = {Finite von {N}eumann algebras and masas}, + SERIES = {London Mathematical Society Lecture Note Series}, + VOLUME = {351}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {2008}, + PAGES = {x+400}, + ISBN = {978-0-521-71919-3}, + MRCLASS = {46L10}, + MRNUMBER = {2433341}, +MRREVIEWER = {Paul\ Jolissaint}, + DOI = {10.1017/CBO9780511666230}, + URL = {https://doi.org/10.1017/CBO9780511666230}, +} + +#Stallings +@article {Stallings65, + AUTHOR = {Stallings, John}, + TITLE = {Homology and central series of groups}, + JOURNAL = {J. Algebra}, + FJOURNAL = {Journal of Algebra}, + VOLUME = {2}, + YEAR = {1965}, + PAGES = {170--181}, + ISSN = {0021-8693}, + MRCLASS = {18.20}, + MRNUMBER = {175956}, +MRREVIEWER = {Hirosi\ Nagao}, + DOI = {10.1016/0021-8693(65)90017-7}, + URL = {https://doi.org/10.1016/0021-8693(65)90017-7}, +} + + +# Sutherland-Takesaki +@article {SutherlandTakesaki89, + AUTHOR = {Sutherland, Colin E. and Takesaki, Masamichi}, + TITLE = {Actions of discrete amenable groups on injective factors of + type {${\rm III}_\lambda,\;\lambda\neq 1$}}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {137}, + YEAR = {1989}, + NUMBER = {2}, + PAGES = {405--444}, + ISSN = {0030-8730,1945-5844}, + MRCLASS = {46L40 (22D25 46L10 46L55)}, + MRNUMBER = {990219}, +MRREVIEWER = {Sze-Kai\ Tsui}, + URL = {http://projecteuclid.org/euclid.pjm/1102650391}, +} + + +# Suziki +@article {Suzuki21, + AUTHOR = {Suzuki, Yuhei}, + TITLE = {Equivariant $\mathcal{O}_2$-absorption theorem for exact groups}, + JOURNAL = {Compos. Math.}, + FJOURNAL = {Compositio Mathematica}, + VOLUME = {157}, + YEAR = {2021}, + NUMBER = {7}, + PAGES = {1492--1506}, + ISSN = {0010-437X,1570-5846}, + MRCLASS = {46L55 (46L05)}, + MRNUMBER = {4275465}, +MRREVIEWER = {Qingzhai\ Fan}, + DOI = {10.1112/s0010437x21007168}, + URL = {https://doi.org/10.1112/s0010437x21007168}, +} + +#Szabo +@article {Szabo18, + AUTHOR = {Szab\'{o}, G\'{a}bor}, + TITLE = {Equivariant {K}irchberg-{P}hillips-type absorption for + amenable group actions}, + JOURNAL = {Comm. Math. Phys.}, + FJOURNAL = {Communications in Mathematical Physics}, + VOLUME = {361}, + YEAR = {2018}, + NUMBER = {3}, + PAGES = {1115--1154}, + ISSN = {0010-3616,1432-0916}, + MRCLASS = {37A55 (46L40)}, + MRNUMBER = {3830263}, +MRREVIEWER = {Francesco\ Fidaleo}, + DOI = {10.1007/s00220-018-3110-3}, + URL = {https://doi.org/10.1007/s00220-018-3110-3}, +} + +### TTTTTTTTTTTTTTTTTTTT + +# Takesaki +@book {TakesakiI, + AUTHOR = {Takesaki, Masamichi}, + TITLE = {Theory of operator algebras. {I}}, + SERIES = {Encyclopaedia of Mathematical Sciences}, + VOLUME = {124}, + NOTE = {Reprint of the first (1979) edition, + Operator Algebras and Non-commutative Geometry, 5}, + PUBLISHER = {Springer-Verlag, Berlin}, + YEAR = {2002}, + PAGES = {xx+415}, + ISBN = {3-540-42248-X}, + MRCLASS = {46Lxx (46-01)}, + MRNUMBER = {1873025}, +} + +@book {TakesakiII, + AUTHOR = {Takesaki, Masamichi}, + TITLE = {Theory of operator algebras. {II}}, + SERIES = {Encyclopaedia of Mathematical Sciences}, + VOLUME = {125}, + NOTE = {Operator Algebras and Non-commutative Geometry, 6}, + PUBLISHER = {Springer-Verlag, Berlin}, + YEAR = {2003}, + PAGES = {xxii+518}, + ISBN = {3-540-42914-X}, + MRCLASS = {46L10 (47L35 47L55)}, + MRNUMBER = {1943006}, +MRREVIEWER = {Robert\ S.\ Doran}, + DOI = {10.1007/978-3-662-10451-4}, + URL = {https://doi.org/10.1007/978-3-662-10451-4}, +} + +@book {TakesakiIII, + AUTHOR = {Takesaki, Masamichi}, + TITLE = {Theory of operator algebras. {III}}, + SERIES = {Encyclopaedia of Mathematical Sciences}, + VOLUME = {127}, + NOTE = {Operator Algebras and Non-commutative Geometry, 8}, + PUBLISHER = {Springer-Verlag, Berlin}, + YEAR = {2003}, + PAGES = {xxii+548}, + ISBN = {3-540-42913-1}, + MRCLASS = {46L10 (37A55 46L35 46L37 46L55)}, + MRNUMBER = {1943007}, +MRREVIEWER = {Robert\ S.\ Doran}, + DOI = {10.1007/978-3-662-10453-8}, + URL = {https://doi.org/10.1007/978-3-662-10453-8}, +} + +# Tatsuuma-Shimomura-Hirai +@article {TatsuumaShimomuraHirai98, + AUTHOR = {Tatsuuma, Nobuhiko and Shimomura, Hiroaki and Hirai, Takeshi}, + TITLE = {On group topologies and unitary representations of inductive + limits of topological groups and the case of the group of + diffeomorphisms}, + JOURNAL = {J. Math. Kyoto Univ.}, + FJOURNAL = {Journal of Mathematics of Kyoto University}, + VOLUME = {38}, + YEAR = {1998}, + NUMBER = {3}, + PAGES = {551--578}, + ISSN = {0023-608X}, + MRCLASS = {22A05 (54H11 58D05)}, + MRNUMBER = {1661157}, +MRREVIEWER = {M. Rajagopalan}, + DOI = {10.1215/kjm/1250518067}, + URL = {https://doi.org/10.1215/kjm/1250518067}, +} + +# Thomsen +@article {Thomsen91, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Nonstable {$K$}-theory for operator algebras}, + JOURNAL = {$K$-Theory}, + FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, + Application, and Influence of $K$-Theory in the Mathematical + Sciences}, + VOLUME = {4}, + YEAR = {1991}, + NUMBER = {3}, + PAGES = {245--267}, + ISSN = {0920-3036}, + MRCLASS = {46L80 (19K33 46L85)}, + MRNUMBER = {1106955}, +MRREVIEWER = {Berndt Brenken}, + DOI = {10.1007/BF00569449}, + URL = {https://doi.org/10.1007/BF00569449}, +} + +@article {Thomsen93, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Finite sums and products of commutators in inductive limit {C}*-algebras}, + JOURNAL = {Ann. Inst. Fourier (Grenoble)}, + FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier}, + VOLUME = {43}, + YEAR = {1993}, + NUMBER = {1}, + PAGES = {225--249}, + ISSN = {0373-0956}, + MRCLASS = {46L05 (46M40)}, + MRNUMBER = {1209702}, +MRREVIEWER = {Guihua Gong}, + URL = {http://www.numdam.org/item?id=AIF_1993__43_1_225_0}, +} + +@article {Thomsen95, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Traces, unitary characters and crossed products by {$\mathbb{Z}$}}, + JOURNAL = {Publ. Res. Inst. Math. Sci.}, + FJOURNAL = {Kyoto University. Research Institute for Mathematical + Sciences. Publications}, + VOLUME = {31}, + YEAR = {1995}, + NUMBER = {6}, + PAGES = {1011--1029}, + ISSN = {0034-5318}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {1382564}, +MRREVIEWER = {Kevin McClanahan}, + DOI = {10.2977/prims/1195163594}, + URL = {https://doi.org/10.2977/prims/1195163594}, +} + + +@article {Thomsen97, + AUTHOR = {Thomsen, Klaus}, + TITLE = {Limits of certain subhomogeneous {C}*-algebras}, + JOURNAL = {M\'{e}m. Soc. Math. Fr. (N.S.)}, + FJOURNAL = {M\'{e}moires de la Soci\'{e}t\'{e} Math\'{e}matique de France. Nouvelle S\'{e}rie}, + NUMBER = {71}, + YEAR = {1997}, + PAGES = {vi+125 pp. (1998)}, + ISSN = {0249-633X}, + MRCLASS = {46L05 (46L80 46M20)}, + MRNUMBER = {1649315}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.24033/msmf.385}, + URL = {https://doi.org/10.24033/msmf.385}, +} + +@misc{Thomsen22, + title={On weights, traces and {K}-theory}, + author={Klaus Thomsen}, + year={2022}, + eprint={2211.03172}, + archivePrefix={arXiv}, + primaryClass={math.OA} +} + +# Tikuisis +@article {Tikuisis12, + AUTHOR = {Tikuisis, Aaron}, + TITLE = {Regularity for stably projectionless, simple {C}*-algebras}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {263}, + YEAR = {2012}, + NUMBER = {5}, + PAGES = {1382--1407}, + ISSN = {0022-1236}, + MRCLASS = {46L35 (46L05)}, + MRNUMBER = {2943734}, +MRREVIEWER = {Hiroyuki Osaka}, + DOI = {10.1016/j.jfa.2012.05.020}, + URL = {https://doi.org/10.1016/j.jfa.2012.05.020}, +} + +@article {Tikuisis16, + AUTHOR = {Tikuisis, Aaron}, + TITLE = {{K}-theoretic characterization of {C}*-algebras with approximately inner flip}, + JOURNAL = {Int. Math. Res. Not. IMRN}, + FJOURNAL = {International Mathematics Research Notices. IMRN}, + YEAR = {2016}, + NUMBER = {18}, + PAGES = {5670--5694}, + ISSN = {1073-7928}, + MRCLASS = {46L35 (19K99)}, + MRNUMBER = {3567256}, +MRREVIEWER = {Cristian Ivanescu}, + DOI = {10.1093/imrn/rnv334}, + URL = {https://doi.org/10.1093/imrn/rnv334}, +} + + +# Tikuisis-White-Winter +@article {TikuisisWhiteWinter17, + AUTHOR = {Tikuisis, Aaron and White, Stuart and Winter, Wilhelm}, + TITLE = {Quasidiagonality of nuclear {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {185}, + YEAR = {2017}, + NUMBER = {1}, + PAGES = {229--284}, + ISSN = {0003-486X}, + MRCLASS = {46L05 (47L40)}, + MRNUMBER = {3583354}, +MRREVIEWER = {Dinesh Jayantilal Karia}, + DOI = {10.4007/annals.2017.185.1.4}, + URL = {https://doi.org/10.4007/annals.2017.185.1.4}, +} + + +# Toms +@article {Toms05, + AUTHOR = {Toms, Andrew S.}, + TITLE = {On the independence of {$K$}-theory and stable rank for simple + {C}*-algebras}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {578}, + YEAR = {2005}, + PAGES = {185--199}, + ISSN = {0075-4102}, + MRCLASS = {46L80 (19K33 46L05)}, + MRNUMBER = {2113894}, +MRREVIEWER = {Nadia S. Larsen}, + DOI = {10.1515/crll.2005.2005.578.185}, + URL = {https://doi.org/10.1515/crll.2005.2005.578.185}, +} + +@article {Toms08, + AUTHOR = {Toms, Andrew S.}, + TITLE = {On the classification problem for nuclear {C}*-algebras}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {167}, + YEAR = {2008}, + NUMBER = {3}, + PAGES = {1029--1044}, + ISSN = {0003-486X}, + MRCLASS = {46L35 (19K14 46L80)}, + MRNUMBER = {2415391}, +MRREVIEWER = {Francesc Perera}, + DOI = {10.4007/annals.2008.167.1029}, + URL = {https://doi.org/10.4007/annals.2008.167.1029}, +} + +@article {Toms08b, + AUTHOR = {Toms, Andrew S.}, + TITLE = {An infinite family of non-isomorphic {C}*-algebras with + identical {$K$}-theory}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {360}, + YEAR = {2008}, + NUMBER = {10}, + PAGES = {5343--5354}, + ISSN = {0002-9947}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {2415076}, +MRREVIEWER = {Wilhelm Winter}, + DOI = {10.1090/S0002-9947-08-04583-2}, + URL = {https://doi.org/10.1090/S0002-9947-08-04583-2}, +} + +# Toms-Winter +@article {TomsWinter07, + AUTHOR = {Toms, Andrew S. and Winter, Wilhelm}, + TITLE = {Strongly self-absorbing {C}*-algebras}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {359}, + YEAR = {2007}, + NUMBER = {8}, + PAGES = {3999--4029}, + ISSN = {0002-9947}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {2302521}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.1090/S0002-9947-07-04173-6}, + URL = {https://doi.org/10.1090/S0002-9947-07-04173-6}, +} + +@article {TomsWinter08, + AUTHOR = {Toms, Andrew S. and Winter, Wilhelm}, + TITLE = {{$\mathcal{Z}$}-stable {ASH} algebras}, + JOURNAL = {Canad. J. Math.}, + FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de + Math\'{e}matiques}, + VOLUME = {60}, + YEAR = {2008}, + NUMBER = {3}, + PAGES = {703--720}, + ISSN = {0008-414X}, + MRCLASS = {46L05 (46L35 46L80)}, + MRNUMBER = {2414961}, +MRREVIEWER = {Hua Xin Lin}, + DOI = {10.4153/CJM-2008-031-6}, + URL = {https://doi.org/10.4153/CJM-2008-031-6}, +} + +@article {TomsWinter09, + AUTHOR = {Toms, Andrew S. and Winter, Wilhelm}, + TITLE = {The {E}lliott conjecture for {V}illadsen algebras of the first + type}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {256}, + YEAR = {2009}, + NUMBER = {5}, + PAGES = {1311--1340}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L35)}, + MRNUMBER = {2490221}, +MRREVIEWER = {Xiao Chun Fang}, + DOI = {10.1016/j.jfa.2008.12.015}, + URL = {https://doi.org/10.1016/j.jfa.2008.12.015}, +} + +# Toms-White-Winer +@article {TomsWhiteWinter15, + AUTHOR = {Toms, Andrew S. and White, Stuart and Winter, Wilhelm}, + TITLE = {$\mathcal{Z}$-stability and finite-dimensional tracial + boundaries}, + JOURNAL = {Int. Math. Res. Not. IMRN}, + FJOURNAL = {International Mathematics Research Notices. IMRN}, + YEAR = {2015}, + NUMBER = {10}, + PAGES = {2702--2727}, + ISSN = {1073-7928}, + MRCLASS = {46L35 (46L40)}, + MRNUMBER = {3352253}, +MRREVIEWER = {C. J. K. Batty}, + DOI = {10.1093/imrn/rnu001}, + URL = {https://doi.org/10.1093/imrn/rnu001}, +} + +### UUUUUUUUUUUUUUUU + +### VVVVVVVVVVVVVVVVV +#Vaes +@article{Vaes24, + title={Every locally compact group is the outer automorphism group of a II$_1$ factor}, + author={Vaes, Stefaan}, + journal={arXiv preprint arXiv:2403.20299}, + year={2024} +} + +# Viladsen +@article {Villadsen98, + AUTHOR = {Villadsen, Jesper}, + TITLE = {Simple {C}*-algebras with perforation}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {154}, + YEAR = {1998}, + NUMBER = {1}, + PAGES = {110--116}, + ISSN = {0022-1236}, + MRCLASS = {46L05 (46L80)}, + MRNUMBER = {1616504}, +MRREVIEWER = {Mahmood Khoshkam}, + DOI = {10.1006/jfan.1997.3168}, + URL = {https://doi.org/10.1006/jfan.1997.3168}, +} + +@article {Villadsen99, + AUTHOR = {Villadsen, Jesper}, + TITLE = {On the stable rank of simple {C}*-algebras}, + JOURNAL = {J. Amer. Math. Soc.}, + FJOURNAL = {Journal of the American Mathematical Society}, + VOLUME = {12}, + YEAR = {1999}, + NUMBER = {4}, + PAGES = {1091--1102}, + ISSN = {0894-0347}, + MRCLASS = {46L05 (18B10 19K99 46L80 46M20)}, + MRNUMBER = {1691013}, +MRREVIEWER = {Vicumpriya S. Perera}, + DOI = {10.1090/S0894-0347-99-00314-8}, + URL = {https://doi.org/10.1090/S0894-0347-99-00314-8}, +} + +# Viola +@article {Viola04, + AUTHOR = {Viola, Maria Grazia}, + TITLE = {On a subfactor construction of a factor non-antisomorphic to + itself}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {15}, + YEAR = {2004}, + NUMBER = {8}, + PAGES = {833--854}, + ISSN = {0129-167X,1793-6519}, + MRCLASS = {46L37 (46L40 46L54)}, + MRNUMBER = {2097021}, +MRREVIEWER = {Teodor\ Banica}, + DOI = {10.1142/S0129167X04002570}, + URL = {https://doi.org/10.1142/S0129167X04002570}, +} + + +# Voiculescu +@article{Voiculescu83, + title={Asymptotically commuting finite rank unitary operators without commuting approximants}, + author={Voiculescu, Dan}, + journal={Acta Sci. Math.(Szeged)}, + volume={45}, + number={1-4}, + pages={429--431}, + year={1983} +} +@article {Voiculescu83, + AUTHOR = {Voiculescu, Dan}, + TITLE = {Asymptotically commuting finite rank unitary operators without + commuting approximants}, + JOURNAL = {Acta Sci. Math. (Szeged)}, + FJOURNAL = {Acta Universitatis Szegediensis. Acta Scientiarum + Mathematicarum}, + VOLUME = {45}, + YEAR = {1983}, + NUMBER = {1-4}, + PAGES = {429--431}, + ISSN = {0001-6969}, + MRCLASS = {47B44}, + MRNUMBER = {708811}, +} + +### WWWWWWWWWWWWWWWWWWWWWWWWWWWW + +# Wagoner +@article {Wagoner72, + AUTHOR = {Wagoner, John B.}, + TITLE = {Delooping classifying spaces in algebraic {K}-theory}, + JOURNAL = {Topology}, + FJOURNAL = {Topology. An International Journal of Mathematics}, + VOLUME = {11}, + YEAR = {1972}, + PAGES = {349--370}, + ISSN = {0040-9383}, + MRCLASS = {18F25 (55B20 55D35)}, + MRNUMBER = {354816}, +MRREVIEWER = {J.\ P.\ May}, + DOI = {10.1016/0040-9383(72)90031-6}, + URL = {https://doi.org/10.1016/0040-9383(72)90031-6}, +} + +# Wen-Fang-Yao +@article{WenFangYao2023, + title={On Diximier's averaging theorem for operators in type {II}$_1$ factors}, + author={Wen, Shilin and Fang, Junsheng and Yao, Zhaolin}, + journal={arXiv preprint arXiv:2303.10602}, + year={2023} +} + +# Wiebel +@book {Kbook, + AUTHOR = {Weibel, Charles A.}, + TITLE = {The {$K$}-book}, + SERIES = {Graduate Studies in Mathematics}, + VOLUME = {145}, + NOTE = {An introduction to algebraic $K$-theory}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {2013}, + PAGES = {xii+618}, + ISBN = {978-0-8218-9132-2}, + MRCLASS = {19-01}, + MRNUMBER = {3076731}, +MRREVIEWER = {L. N. Vaserstein}, + DOI = {10.1090/gsm/145}, + URL = {https://doi.org/10.1090/gsm/145}, +} + +# Willett-Yu +@article{WillettYu21, + title={The {UCT} for {C}*-algebras with finite complexity}, + author={Willett, Rufus and Yu, Guoliang}, + journal={arXiv e-prints}, + pages={arXiv--2104}, + year={2021} +} + +# Winter +@article {Winter11, + AUTHOR = {Winter, Wilhelm}, + TITLE = {Strongly self-absorbing {C}*-algebras are $\mathcal{Z}$-stable}, + JOURNAL = {J. Noncommut. Geom.}, + FJOURNAL = {Journal of Noncommutative Geometry}, + VOLUME = {5}, + YEAR = {2011}, + NUMBER = {2}, + PAGES = {253--264}, + ISSN = {1661-6952}, + MRCLASS = {46L35 (46L05 46L80)}, + MRNUMBER = {2784504}, +MRREVIEWER = {Camillo Trapani}, + DOI = {10.4171/JNCG/74}, + URL = {https://doi.org/10.4171/JNCG/74}, +} + +@article {Winter12, + AUTHOR = {Winter, Wilhelm}, + TITLE = {Nuclear dimension and $\mathcal{Z}$-stability of pure {C}*-algebras}, + JOURNAL = {Invent. Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {187}, + YEAR = {2012}, + NUMBER = {2}, + PAGES = {259--342}, + ISSN = {0020-9910}, + MRCLASS = {46L85 (46L35)}, + MRNUMBER = {2885621}, + DOI = {10.1007/s00222-011-0334-7}, + URL = {https://doi.org/10.1007/s00222-011-0334-7}, +} + +### XXXXXXXXXXXXXXXXXXXXX + +### YYYYYYYYYYYYYYYYYYYYY + +# Yen +@article {Yen56, + AUTHOR = {Yen, Ti}, + TITLE = {Isomorphism of unitary groups in {AW}*-algebras}, + JOURNAL = {Tohoku Math. J. (2)}, + FJOURNAL = {The Tohoku Mathematical Journal. Second Series}, + VOLUME = {8}, + YEAR = {1956}, + PAGES = {275--280}, + ISSN = {0040-8735}, + MRCLASS = {46.2X}, + MRNUMBER = {89376}, +MRREVIEWER = {I. E. Segal}, + DOI = {10.2748/tmj/1178244951}, + URL = {https://doi.org/10.2748/tmj/1178244951}, +} + +### ZZZZZZZZZZZZZZZZZZZZ + + diff --git a/Universal covering groups of unitary groups of von Neumann algebras/letterfonts.tex b/Universal covering groups of unitary groups of von Neumann algebras/letterfonts.tex new file mode 100644 index 0000000..449ad2a --- /dev/null +++ b/Universal covering groups of unitary groups of von Neumann algebras/letterfonts.tex @@ -0,0 +1,90 @@ +%commands +\newcommand{\ms}{\mathscr} +\newcommand{\mc}{\mathcal} +\newcommand{\mf}{\mathfrak} +\newcommand{\nin}{\not\in} +\newcommand{\bs}{\backslash} +\newcommand{\nsg}{\unlhd} +\newcommand{\ov}{\overline} +\newcommand{\wt}{\widetilde} +%blackboard +\newcommand{\bN}{\mathbb{N}} +\newcommand{\bR}{\mathbb{R}} +\newcommand{\bZ}{\mathbb{Z}} +\newcommand{\bQ}{\mathbb{Q}} +\newcommand{\bF}{\mathbb{F}} +\newcommand{\bK}{\mathbb{K}} +\newcommand{\bG}{\mathbb{G}} +\newcommand{\bE}{\mathbb{E}} +\newcommand{\bC}{\mathbb{C}} +\newcommand{\bV}{\mathbb{V}} +\newcommand{\bA}{\mathbb{A}} +\newcommand{\bP}{\mathbb{P}} +\newcommand{\bD}{\mathbb{D}} +\newcommand{\bT}{\mathbb{T}} +\newcommand{\bM}{\mathbb{M}} +\newcommand{\bB}{\mathbb{B}} +\newcommand{\bL}{\mathbb{L}} +\newcommand{\bX}{\mathbb{X}} +\newcommand{\bY}{\mathbb{Y}} +\newcommand{\bI}{\mathbb{I}} + + +\newcommand{\1}{{\bf 1}} +\newcommand{\0}{{\bf 0}} + +%mathcal +\newcommand{\cM}{\mathcal{M}} +\newcommand{\cN}{\mathcal{N}} +\newcommand{\cC}{\mathcal{C}} +\newcommand{\cB}{\mathcal{B}} +\newcommand{\cS}{\mathcal{S}} +\newcommand{\cI}{\mathcal{I}} +\newcommand{\cO}{\mathcal{O}} +\newcommand{\cP}{\mathcal{P}} +\newcommand{\cA}{\mathcal{A}} +\newcommand{\cL}{\mathcal{L}} +\newcommand{\cF}{\mathcal{F}} +\newcommand{\cH}{\mathcal{H}} +\newcommand{\cK}{\mathcal{K}} +\newcommand{\cD}{\mathcal{D}} +\newcommand{\cE}{\mathcal{E}} +\newcommand{\cT}{\mathcal{T}} +\newcommand{\cZ}{\mathcal{Z}} +\newcommand{\cU}{\mathcal{U}} +\newcommand{\cJ}{\mathcal{J}} +\newcommand{\cG}{\mathcal{G}} +\newcommand{\cR}{\mathcal{R}} +\newcommand{\cQ}{\mathcal{Q}} +\newcommand{\cY}{\mathcal{Y}} +\newcommand{\cW}{\mathcal{W}} +%terns +\newcommand{\bh}{\mathcal{B}(\mathcal{H})} +\newcommand{\bk}{\mathcal{B}(\mathcal{K})} +\newcommand{\kh}{\mathcal{K}(\mathcal{H})} +\newcommand{\fh}{\mathcal{F}(\mathcal{H})} +\newcommand{\sa}{\mathcal{S}(\mathcal{A})} +\newcommand{\bx}{\mathcal{B}(\mathbb{X}} +\newcommand{\by}{\mathcal{C}(\mathbb{Y}} + +%scr +\newcommand{\scH}{\mathscr{H}} +\newcommand{\scG}{\mathscr{G}} +\newcommand{\scE}{\mathscr{E}} + +%frak +\newcommand{\fX}{\mathfrak{X}} +\newcommand{\fY}{\mathfrak{Y}} +\newcommand{\fS}{\mathfrak{S}} +\newcommand{\fM}{\mathfrak{M}} +\newcommand{\fN}{\mathfrak{N}} +\newcommand{\fF}{\mathfrak{F}} +\newcommand{\fJ}{\mathfrak{J}} +\newcommand{\fT}{\mathfrak{T}} +\newcommand{\fA}{\mathfrak{A}} +\newcommand{\fB}{\mathfrak{B}} +\newcommand{\fP}{\mathfrak{P}} +\newcommand{\fQ}{\mathfrak{Q}} +\newcommand{\fO}{\mathfrak{O}} +\newcommand{\fK}{\mathfrak{K}} +\newcommand{\fG}{\mathfrak{G}} diff --git a/Universal covering groups of unitary groups of von Neumann algebras/macros.tex b/Universal covering groups of unitary groups of von Neumann algebras/macros.tex new file mode 100644 index 0000000..c0edaa1 --- /dev/null +++ b/Universal covering groups of unitary groups of von Neumann algebras/macros.tex @@ -0,0 +1,143 @@ +% tensor products +\newcommand{\thk}{\cH\otimes\cK} +\newcommand{\amb}{\bA \otimes_{\max} \bB} +\newcommand{\omax}{\otimes_{\max}} +\newcommand{\omin}{\otimes_{\min}} +\newcommand{\ovn}{\overline{\otimes}} + +% probability +\newcommand{\PS}{(\Omega,\cM,\cP)} +\newcommand{\rvf}{f:\Omega \to \bR} +\newcommand{\linf}{\cL^{\infty -}\PS} +\newcommand{\probc}{\text{Prob}_c(\bR)} +\newcommand{\prob}{\text{Prob}(\bR)} +\newcommand{\Prob}{\text{Prob}} +\newcommand{\Bor}{\text{Bor}} + +% maps +\newcommand{\into}{\hookrightarrow} +\newcommand{\onto}{\twoheadrightarrow} +\newcommand{\act}{\curvearrowright} + +\newcommand{\ee}{\varepsilon} +\newcommand{\sm}{\setminus} +\newcommand{\Om}{\Omega} +\newcommand{\simi}{\sim_{\infty}} + +%limits +\newcommand{\limsupn}{\underset{n}{\text{limsup}}} +\newcommand{\sotc}{\stackrel{\text{SOT}}{\to}} +\newcommand{\wotc}{\stackrel{\text{WOT}}{\to}} +\newcommand{\sotlim}{\text{SOT-}\lim} +\newcommand{\wotlim}{\text{WOT-}\lim} + +% useful for nuclear, lifting maps, etc.... +\newcommand{\tphi}{\tilde{\phi}} +\newcommand{\tpsi}{\tilde{\psi}} +\newcommand{\hphi}{\hat{\phi}} +\newcommand{\hpsi}{\hat{\psi}} +\newcommand{\phil}{\phi_\lambda} +\newcommand{\psil}{\psi_\lambda} +\newcommand{\mkl}{M_{k(\lambda)}} +\newcommand{\mkn}{M_{k(n)}} +\newcommand{\phin}{\phi_n} +\newcommand{\psin}{\psi_n} +\newcommand{\vphi}{\varphi} + +\newcommand{\floor}[1]{\lfloor #1 \rfloor} + +\newcommand{\supp}{\text{supp}} +\newcommand{\wk}{\text{wk}} +\newcommand{\Tr}{\text{Tr}} +\newcommand{\dist}{\text{dist}} +\newcommand{\rank}{\text{rank}} +\newcommand{\sgn}{\text{sgn}} + +\newcommand{\ev}{\text{ev}} +\newcommand{\spr}{\text{spr}} +\newcommand{\tD}{\tilde{\Delta}} +\newcommand{\uv}{\underline} +\newcommand{\tr}{\text{tr}} +\newcommand{\id}{\text{id}} +\newcommand{\ad}{\text{ad}} +\newcommand{\Ad}{\text{Ad}} +\newcommand{\Inn}{\text{Inn}} +\newcommand{\Out}{\text{Out}} +\newcommand{\Ct}{\text{Ct}} + +\newcommand*{\tc}[2]{(\ov{#1}^{#2},#2)} + +% Types for von Neumann algebras +\newcommand{\I}{\text{I}} +\newcommand{\II}{\text{II}} +\newcommand{\III}{\text{III}} +\newcommand{\IIIl}{\text{III}_{\lambda}} +% note the last character of this last one is an "ell", its just that my font has capital i and lower-case l being the same character..... + + +%operators +% basic algebra +\DeclareMathOperator{\Ann}{Ann} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\im}{im} + +%Grassmanian +\DeclareMathOperator{\Gr}{Gr} + +% K-theory and regularity for C*-algebras +\DeclareMathOperator{\Ell}{Ell} +\DeclareMathOperator{\Aff}{Aff} +%\DeclareMathOperator{\KT}{KT} +\newcommand{\KTu}{\underline{K}T_u} +\DeclareMathOperator{\Cu}{Cu} +%the following was used in ``polar decomp. in alg. K-theory'' with Aaron. +%\DeclareMathOperator{\ka}{K_1^{\text{alg}}} +%\DeclareMathOperator{\ku}{K_1^{\text{alg,u}}} +%\DeclareMathOperator{\hka}{\ov{K}_1^{\text{alg}}} +%\DeclareMathOperator{\hku}{\ov{K}_1^{\text{alg,u}}} +\DeclareMathOperator{\St}{\text{St}} +%topological K-theory +\newcommand*{\ktop}[1]{K^{\text{top}}_{#1}} +%unitary algebraic K-theory +\newcommand*{\kua}[1]{K^{\text{alg,u}}_{#1}} +%algebraic K-theory +\newcommand*{\ka}[1]{K^{\text{alg}}_{#1}} +%hausdorffized variants +\newcommand*{\hkua}[1]{\ov{K}^{\text{alg,u}}_{#1}} +\newcommand*{\hka}[1]{\ov{K}^{\text{alg}}_{#1}} + +\newcommand{\MvN}{\sim_{\text{MvN}}} + + +% exponential rank and length for C*-algebras +\newcommand{\cer}{\text{cer}} +\newcommand{\cel}{\text{cel}} + +% KK-theory and Ext +\DeclareMathOperator{\KK}{KK} +\DeclareMathOperator{\KKn}{KK_{\text{nuc}}} +\DeclareMathOperator{\KKs}{KK_{\text{sep}}} +\DeclareMathOperator{\Ext}{Ext} +\DeclareMathOperator{\Extn}{Ext_{\text{nuc}}} + +% direct/inverse limits +\newcommand{\dlim}{\underset{\to}{\lim}} +\newcommand{\ilim}{\underset{\leftarrow}{\lim}} + + +% group homology +\newcommand{\Tor}{\text{Tor}} + +% dynamics +\newcommand{\Fix}{\text{Fix}} + +% ideal stuff? +\newcommand{\Ped}{\text{Ped}} + +% Brown algebra stuff +\newcommand{\Unc}{\mathcal{U}_{\text{nc}}} + +% C*-dynamics +\newcommand{\RUC}{\text{RUC}} % right uniformly continuous diff --git a/Universal covering groups of unitary groups of von Neumann algebras/preabmle.tex b/Universal covering groups of unitary groups of von Neumann algebras/preabmle.tex new file mode 100644 index 0000000..4288b08 --- /dev/null +++ b/Universal covering groups of unitary groups of von Neumann algebras/preabmle.tex @@ -0,0 +1,57 @@ +% change this depending on what you're doing. +%\documentclass[11pt]{amsart} + +%packages +%\usepackage[margin=1.5in]{geometry} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[pdftex]{graphicx} +\usepackage{tikz-cd} +\usepackage{blindtext} + +\usepackage{hyperref,xcolor} +\usepackage{amsmath,amsfonts,amssymb,amsthm,color,mathtools,enumitem} +\usepackage{epstopdf} +\usepackage{polynom} +\usepackage{babel} +\usepackage{mathrsfs} +\usepackage{chngcntr} +\usepackage{verbatim} + +\hypersetup{ + colorlinks=true, + linkcolor=blue, + citecolor=cyan, + urlcolor=cyan} + + \usepackage{hyphenat} + + %theorems +\newtheorem{theorem}{Theorem}[section] % for normal paper +%\newtheorem{theorem}{Theorem} for short paper +\newtheorem{defn}[theorem]{Definition} +\newtheorem{prop}[theorem]{Proposition} +\newtheorem{cor}[theorem]{Corollary} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{example}[theorem]{Example} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{question}[theorem]{Question} +\newtheorem{conj}[theorem]{Conjecture} + +\newtheorem*{resultA}{Theorem A} +\newtheorem*{resultB}{Theorem B} +\newtheorem*{resultC}{Theorem C} +\newtheorem*{resultD}{Theorem D} +\newtheorem*{resultE}{Theorem E} +\newtheorem*{resultF}{Theorem F} + +\newtheorem*{result*}{Theorem} +\newtheorem*{resultcor*}{Corollary} +\newtheorem{result}{Theorem} +%\newtheorem{result}{Theorem} for short paper +\renewcommand*{\theresult}{\arabic{chapter}.\Alph{result}} +\renewcommand*{\theresult}{\Alph{result}} +\newtheorem{resultcor}[result]{Corollary} + +\numberwithin{equation}{section} +