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Nuclear C*-algebra

In the mathematical field of functional analysis, a nuclear C-algebra* is a C*-algebra A such that for every C*-algebra B the injective and projective C*-cross norms coincides on the algebraic tensor product A⊗B and the completion of A⊗B 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 B of the Cuntz algebra 𝒪2 with the property that there exists a conditional expectation from 𝒪2 to B.

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 n×n 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.

References

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

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

c*-algebras functional-analysis operator-theory
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