% 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}}