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 Mn 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 Mn, and also a linear isomorphism T : E → F such that the cb-norm
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
