Definition 0.1   A C*-algebra is called a nuclear C*-algebra if all C*-norms on every algebraic tensor product , of with any other C*-algebra , agree with, and also equal the spatial C*-norm (viz Lance, 1981). Therefore, there is a unique completion of to a C*-algebra , for any other C*-algebra .

Examples of nuclear -algebras: All commutative C*-algebras and all finite-dimensional C*-algebras .

Other examples* of nuclear C*-algebras :

• group C*-algebras of amenable groups,
• crossed products of strongly amenable C*-algebras by amenable discrete groups,
• Type C*-algebras.

Exact -algebra In general terms, a -algebra is exact if it is isomorphic with a -subalgebra of some nuclear -algebra. The precise definition of an exact -algebra follows.

Definition 0.2   Let be a matrix space, let be a general operator space, and also let be a C*-algebra. A -algebra is exact if it is `finitely representable' in , that is, if for every finite dimensional subspace in and quantity , there exists a subspace of some , and also a linear isomorphism such that the -norm

.

Counter-Example: The group C*-algebras for the free groups on two or more generators are not nuclear. Furthermore, a -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 -algebras, Hilbert -Modules and Quantum Mechanics", pp. 89 a graduate level preprint discussing general C*-algebras in Postscript format.

*contributed by two PM members