Girish Mahajan (Editor)

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 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.

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.
  • It is isomorphic to a C*-subalgebra B of the Cuntz algebra O 2 with the property that there exists a conditional expectation from O 2 to B. This condition is only equivalent to the others for separable C*-algebras.

    • References

    Nuclear C*-algebra Wikipedia
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