Nuclear C*-algebra explained

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.
• (For separable algebras) 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 (real or complex-valued) continuous functions on a compact Hausdorff space as well as the noncommutative unital algebra of real or complex matrices are nuclear.^[1]

Notes and References

1. Argerami, Martin (20 January 2023). Answer to "The C∗ algebras of matrices, continuous functions, measures and matrix-valued measures / continuous functions and their state spaces". Mathematics StackExchange. Stack Exchange.