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.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.
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References
it:C*-algebra#C*-algebra nucleare