PlanetPhysics/Nuclear C Algebra

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

 A C*-algebra $A$ is called a 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 (viz  Lance, 1981). Therefore, there is a unique completion of $A\otimes X$ to a C*-algebra , for any other C*-algebra $X$.

Examples of nuclear C*-algebras

• All commutative C*-algebras and all finite-dimensional C*-algebras
• group C*-algebras of amenable groups
• Crossed products of strongly amenable C*-algebras by amenable discrete groups,
• type $1$ C*-algebras.

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 exact $C^{*}$-algebra follows.

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 $|T|_{cb}|T^{-1}|_{cb}<1+epsilon.$

Counter-example

The group C*-algebras for the free groups on two or more generators are not nuclear. Furthermore, a $C^{*}$ -subalgebra of a nuclear C*-algebra need not be nuclear.