Nuclear C*-algebra

In the mathematical field of functional analysis, a nuclear C*-algebra is a C*-algebra such that for every C*-algebra the injective and projective C*-cross norms coincides on the algebraic tensor product and the completion of with respect to this norm is a C*-algebra. This property was first studied by under the name "Property T", which is not related to Kazhdan's property T.

Characterizations

Nuclearity admits the following equivalent characterizations:

The identity map, as a completely positive map, approximately factors through matrix algebras. By this equivalence, nuclearity can be considered a noncommutative analogue of the existence of partitions of unity.
The enveloping von Neumann algebra is injective.
It is amenable as a Banach algebra.
It is isomorphic to a C*-subalgebra of the Cuntz algebra with the property that there exists a conditional expectation from to.

Examples

The commutative unital C* algebra of continuous functions on a compact Hausdorff space as well as the noncommutative unital algebra of real or complex matrices are nuclear.