nuclear C*-algebra

Definition 0.1.

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 $X$.

0.1 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.

0.2 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 $C^{\mathrm{*}}$-algebra follows.

Definition 0.2.

Let $M_n$ be a matrix space, let $𝒜$ be a general operator space, and also let $ℂ$ be a C*-algebra.A $C^{*}$-algebra $ℂ$ is exact if it is ‘finitely representable’ in $M_n$, that is, if for every finite dimensional subspace $E$ in $𝒜$ and quantity $epsilon>0$, there exists a subspace $F$ of some $M_n$, andalso a linear isomorphism $T:E\to F$ such that the $cb$-norm

0.3 Note: A counter-example

A $C^{*}$ -subalgebra of a nuclear C*-algebra need not be nuclear.