Nuclear C*-algebra
Introduction 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.
See also * * * *
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