Nuclear C*-algebra

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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 AB are the same for every C*-algebra B. These were first studied by Takesaki (1964) under the name "Property T" (this is unconnected with Kazhdan's property T).


Nuclearity admits the following equivalent characterizations:

  • A C*-algebra is nuclear if the identity map, as a completely positive map, approximately factors through matrix algebras. Abelian C*-algebras are nuclear. One might say that nuclearity means that the C*-algebra admits noncommutative "partitions of unity."
  • A C*-algebra is nuclear if and only if its enveloping von Neumann algebra is injective.
  • A separable C*-algebra is nuclear if and only if it is isomorphic to a C*-subalgebra B of the Cuntz algebra \mathcal{O}_2 with the property that there exists a conditional expectation from \mathcal{O}_2 to B.
  • A C*-algebra is nuclear if and only if it is amenable as a Banach algebra.

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