PlanetPhysics/Nuclear C Algebra

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 A [[../VonNeumannAlgebra2/|C*-algebra]] $A$ is called a nuclear  C*-algebra if all C*-norms on every [[../CoIntersections/|algebraic]] [[../Tensor/|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
• [[../TrivialGroupoid/|group]] C*-algebras of amenable groups
• Crossed products of strongly amenable C*-algebras by amenable discrete groups,
• [[../Bijective/|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/|matrix]] space, let $𝒜$ be a general [[../QuantumSpinNetworkFunctor2/|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 $M_n$, and also a linear [[../IsomorphicObjectsUnderAnIsomorphism/|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 [[../Generator/|generators]] are not nuclear. Furthermore, a $C^{*}$ -subalgebra of a nuclear C*-algebra need not be nuclear.