TeX files

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