Jump to content

Talk:PlanetPhysics/Nuclear C Algebra

Page contents not supported in other languages.
Add topic
From Wikiversity

Original TeX Content from PlanetPhysics Archive

[edit source]
%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: nuclear C*-algebra
%%% Primary Category Code: 00.
%%% Filename: NuclearCAlgebra.tex
%%% Version: 1
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.        
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetMath preamble. as your knowledge

% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

% define commands here
\usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate}
\usepackage{xypic, xspace}
\usepackage[mathscr]{eucal}
\usepackage[dvips]{graphicx}
\usepackage[curve]{xy}

\setlength{\textwidth}{6.5in}
%\setlength{\textwidth}{16cm}
\setlength{\textheight}{9.0in}
%\setlength{\textheight}{24cm}

\hoffset=-.75in %%ps format
%\hoffset=-1.0in %%hp format
\voffset=-.4in


\theoremstyle{plain}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]

\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]
%\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\newtheorem*{notation}{Notation}
\newtheorem*{claim}{Claim}

\renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}}
\numberwithin{equation}{section}

\newcommand{\Ad}{{\rm Ad}}
\newcommand{\Aut}{{\rm Aut}}
\newcommand{\Cl}{{\rm Cl}}
\newcommand{\Co}{{\rm Co}}
\newcommand{\DES}{{\rm DES}}
\newcommand{\Diff}{{\rm Diff}}
\newcommand{\Dom}{{\rm Dom}}
\newcommand{\Hol}{{\rm Hol}}
\newcommand{\Mon}{{\rm Mon}}
\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Ker}{{\rm Ker}}
\newcommand{\Ind}{{\rm Ind}}
\newcommand{\IM}{{\rm Im}}
\newcommand{\Is}{{\rm Is}}
\newcommand{\ID}{{\rm id}}
\newcommand{\grpL}{{\rm GL}}
\newcommand{\Iso}{{\rm Iso}}
\newcommand{\rO}{{\rm O}}
\newcommand{\Sem}{{\rm Sem}}
\newcommand{\SL}{{\rm Sl}}
\newcommand{\St}{{\rm St}}
\newcommand{\Sym}{{\rm Sym}}
\newcommand{\Symb}{{\rm Symb}}
\newcommand{\SU}{{\rm SU}}
\newcommand{\Tor}{{\rm Tor}}
\newcommand{\U}{{\rm U}}

\newcommand{\A}{\mathcal A}
\newcommand{\Ce}{\mathcal C}
\newcommand{\D}{\mathcal D}
\newcommand{\E}{\mathcal E}
\newcommand{\F}{\mathcal F}
%\newcommand{\grp}{\mathcal G}
\renewcommand{\H}{\mathcal H}
\renewcommand{\cL}{\mathcal L}
\newcommand{\Q}{\mathcal Q}
\newcommand{\R}{\mathcal R}
\newcommand{\cS}{\mathcal S}
\newcommand{\cU}{\mathcal U}
\newcommand{\W}{\mathcal W}

\newcommand{\bA}{\mathbb{A}}
\newcommand{\bB}{\mathbb{B}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bG}{\mathbb{G}}
\newcommand{\bK}{\mathbb{K}}
\newcommand{\bM}{\mathbb{M}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bO}{\mathbb{O}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bV}{\mathbb{V}}
\newcommand{\bZ}{\mathbb{Z}}

\newcommand{\bfE}{\mathbf{E}}
\newcommand{\bfX}{\mathbf{X}}
\newcommand{\bfY}{\mathbf{Y}}
\newcommand{\bfZ}{\mathbf{Z}}

\renewcommand{\O}{\Omega}
\renewcommand{\o}{\omega}
\newcommand{\vp}{\varphi}
\newcommand{\vep}{\varepsilon}

\newcommand{\diag}{{\rm diag}}
\newcommand{\grp}{{\mathsf{G}}}
\newcommand{\dgrp}{{\mathsf{D}}}
\newcommand{\desp}{{\mathsf{D}^{\rm{es}}}}
\newcommand{\grpeod}{{\rm Geod}}
%\newcommand{\grpeod}{{\rm geod}}
\newcommand{\hgr}{{\mathsf{H}}}
\newcommand{\mgr}{{\mathsf{M}}}
\newcommand{\ob}{{\rm Ob}}
\newcommand{\obg}{{\rm Ob(\mathsf{G)}}}
\newcommand{\obgp}{{\rm Ob(\mathsf{G}')}}
\newcommand{\obh}{{\rm Ob(\mathsf{H})}}
\newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}}
\newcommand{\grphomotop}{{\rho_2^{\square}}}
\newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}}

\newcommand{\rf}{{R_{\mathcal F}}}
\newcommand{\grplob}{{\rm glob}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\TOP}{{\rm TOP}}
\newcommand{\wti}{\widetilde}
\newcommand{\what}{\widehat}
\renewcommand{\a}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\grpa}{\grpamma}
%\newcommand{\grpa}{\grpamma}
\newcommand{\de}{\delta}
\newcommand{\del}{\partial}
\newcommand{\ka}{\kappa}
\newcommand{\si}{\sigma}
\newcommand{\ta}{\tau}
\newcommand{\lra}{{\longrightarrow}}
\newcommand{\ra}{{\rightarrow}}
\newcommand{\rat}{{\rightarrowtail}}
\newcommand{\ovset}[1]{\overset {#1}{\ra}}
\newcommand{\ovsetl}[1]{\overset {#1}{\lra}}
\newcommand{\hr}{{\hookrightarrow}}

\newcommand{\<}{{\langle}}

\def\baselinestretch{1.1}

\hyphenation{prod-ucts}

%\grpeometry{textwidth= 16 cm, textheight=21 cm}

\newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}&
#3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram
\eqno{\mbox{#9}}$$ }

\def\C{C^{\ast}}

\newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}}

%\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$
%{\mbox{}}
\newcommand{\quadr}[4]
{\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4&
\end{pmatrix}}
\def\D{\mathsf{D}}

\begin{document}

 \begin{definition} A \htmladdnormallink{C*-algebra}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html} $A$ is called a {\em nuclear} C*-algebra if all C*-norms on every \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html} product $A \otimes X$, of $A$ with any other C*-algebra $X$, agree with, and also equal the spatial C*-norm (\emph{viz} Lance, 1981). Therefore, there is a unique completion of $A \otimes X$ to a C*-algebra , for any other C*-algebra $X$.
\end{definition}

\subsection{Examples of nuclear C*-algebras}

\begin{itemize}
\item All commutative C*-algebras and all finite-dimensional C*-algebras
\item \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} C*-algebras of amenable groups
\item Crossed products of strongly amenable C*-algebras by amenable discrete groups,
\item \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} $1$ C*-algebras.
\end{itemize}

\subsection{Exact C*-algebra}
In general terms, a $C^*$-algebra is exact if it is isomorphic with a $C^*$-subalgebra of some nuclear $C^*$-algebra. The precise definition of an \emph{exact $C^*$-algebra} follows.

\begin{definition}
Let $M_n$ be a \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} space, let $\mathcal{A}$ be a general \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} space, and also let $\mathbb{C}$ be a C*-algebra.
A $C^*$-algebra $\mathbb{C}$ is exact if it is `finitely representable' in $M_n$, that is, if for every finite dimensional subspace $E$ in $\mathcal{A}$ and quantity $epsilon > 0$, there exists a subspace $F$ of some $M_n$, and
also a linear \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} $T:E \to F$ such that the $cb$-norm
$$|T|_{cb}|T^{-1}|_{cb} < 1 + epsilon.$$
\end{definition}

\subsection{Counter-example}
The group C*-algebras for the free groups on two or more \htmladdnormallink{generators}{http://planetphysics.us/encyclopedia/Generator.html} are not nuclear.
Furthermore, a $C^*$ -subalgebra of a nuclear C*-algebra \textbf{need not be} nuclear.

\begin{thebibliography}{9}
\bibitem{LEC81}
E. C. Lance. 1981. Tensor Products and nuclear C*-algebras., in {\em Operator
Algebras and Applications,} R.V. Kadison, ed., Proceed. Symp. Pure Maths., \textbf{38}: 379-399, part 1.

\bibitem{LN98}
N. P. Landsman. 1998. ``Lecture notes on $C^*$-algebras, Hilbert $C^*$-Modules and Quantum Mechanics'', pp. 89
\htmladdnormallink{a graduate level preprint discussing general C*-algebras}{http://planetmath.org/?op=getobj&from=books&id=66}
\htmladdnormallink{in Postscript format}{http://aux.planetmath.org/files/books/66/C*algebrae.ps}.

\end{thebibliography} 

\end{document}