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.