nuclear C*-algebra NuclearCAlgebra 2008-07-14 03:32:39 2009-02-02 13:46:37 Definition generated C*-algebra exact C^*-algebra quantum operator algebra nuclear C*-algebra generators % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} 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%\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{definition} A C*-algebra $A$ is called a {\em nuclear} C*-algebra if all C*-norms on every algebraic tensor 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 Group C*-algebras of amenable groups \item Crossed products of strongly amenable C*-algebras by amenable discrete groups, \item Type $1$ 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 matrix space, let $\mathcal{A}$ be a general operator 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 isomorphism $T:E \to F$ such that the $cb$-norm $$|T|_{cb}|T^{-1}|_{cb} < 1 + epsilon.$$ \end{definition} \subsection{Note: A counter-example} 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 \PMlinkexternal{a graduate level preprint discussing general C*-algebras}{http://planetmath.org/?op=getobj&from=books&id=66} \PMlinkexternal{in Postscript format}{http://aux.planetmath.org/files/books/66/C*algebrae.ps}. \end{thebibliography}