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$ .

Examples of nuclear 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.

Definition 0.2   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

$\displaystyle \vert T\vert _{cb}\vert T^{-1}\vert _{cb} < 1 + epsilon.$

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.

Bibliography

1
E. C. Lance. 1981. Tensor Products and nuclear C*-algebras., in Operator Algebras and Applications, R.V. Kadison, ed., Proceed. Symp. Pure Maths., 38: 379-399, part 1.

2
N. P. Landsman. 1998. ``Lecture notes on $ C^*$ -algebras, Hilbert $ C^*$ -Modules and Quantum Mechanics'', pp. 89 a graduate level preprint discussing general C*-algebras in Postscript format.



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