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nuclear C*-algebra (Definition)
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

  • 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.
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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Keywords:  nuclear C*-algebras

Cross-references: generators, isomorphism, operator, matrix, type, group, tensor, algebraic, C*-algebra

This is version 1 of nuclear C*-algebra, born on 2009-02-02.
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Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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