# PlanetPhysics/Nuclear C Algebra

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 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.