81T05-NuclearCalgebra.tex

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\pmtitle{nuclear C*-algebra}
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\pmclassification{msc}{81T05}
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\pmdefines{generated C*-algebra}
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\pmdefines{group C*-algebra}

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\begin{document}
\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}


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