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In mathematics, a nuclear C*-algebra is a C*-algebra A such that the injective and projective C*-cross norms on A⊗B are the same for every C*-algebra B. This property was first studied by Takesaki (1964) under the name "Property T", which is not related to Kazhdan's property T.
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 B of the Cuntz algebra with the property that there exists a conditional expectation from to B. This condition is only equivalent to the others for separable C*-algebras.
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