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Unitary groups, K-theory and traces/biblio.bib Normal file
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Unitary groups, K-theory and traces/letterfonts.tex Normal file
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%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}}

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Unitary groups, K-theory and traces/macros.tex Normal file
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% 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}}

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% 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}

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\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}
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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,
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John~B. Conway, \emph{A course in functional analysis}, Graduate Texts in
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Joachim Cuntz and Gert~K. Pedersen, \emph{Equivalence and traces on
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simple amenable $\mathcal{Z}$-stable {C}*-algebras, {I}: {C}*-algebras with
generalized tracial rank one}, C. R. Math. Acad. Sci. Soc. R. Can.
\textbf{42} (2020), no.~3, 63--450. \MR{4215379}
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no.~1, 41--55, 56--100. \MR{1466905}
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\bibitem[NR17]{NgRobert17}
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projections in a {C}*-algebra}, Internat. J. Math. \textbf{3} (1992), no.~5,
679--698. \MR{1189681}
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\end{thebibliography}

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\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}%