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%%% Primary Title: nuclear C*-algebra
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\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}
