nuclear C*-algebra | Physics Library

Definition 0.1. A C*-algebra A is called a nuclear C*-algebra if all C*-norms on every algebraic tensor product A⊗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 ⊗ X to a C*-algebra , for any other C*-algebra X.

0.1 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.

0.2 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 𝒜 be a general operator space, and also let ℂ be a C*-algebra. A C^∗-algebra ℂ is exact if it is ‘finitely representable’ in M_n, that is, if for every finite dimensional subspace E in 𝒜 and quantity epsilon > 0, there exists a subspace F of some M_n, and also a linear isomorphism T : E → F such that the cb-norm

−1 |T |cb|T |cb < 1 + epsilon.

0.3 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.

References

[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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