# PlanetPhysics/Nuclear C Algebra

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A C*-algebra is called anuclearC*-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 (vizLance, 1981). Therefore, there is a unique completion of to a C*-algebra , for any other C*-algebra .

### Examples of nuclear C*-algebrasEdit

- 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 C*-algebras.

### Exact C*-algebraEdit

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.

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-exampleEdit

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.

## All SourcesEdit

^{[1]}^{[2]}

## ReferencesEdit

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