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Exact C*-algebra

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In mathematics, an exact C*-algebra is a C*-algebra that preserves exact sequences under the minimum tensor product.

Definition

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A C*-algebra E is exact if, for any short exact sequence,

the sequence

where ⊗min denotes the minimum tensor product, is also exact.

Properties

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  • Every sub-C*-algebra and every quotient of an exact C*-algebra is exact. An extension of exact C*-algebras is not exact in general.

Characterizations

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Exact C*-algebras have the following equivalent characterizations:

  • A C*-algebra A is exact if and only if A is nuclearly embeddable into B(H), the C*-algebra of all bounded operators on a Hilbert space H.
  • A C*-algebra is exact if and only if every separable sub-C*-algebra is exact.
  • A separable C*-algebra A is exact if and only if it is isomorphic to a subalgebra of the Cuntz algebra .

References

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  • Brown, Nathanial P.; Ozawa, Narutaka (2008). C*-algebras and Finite-Dimensional Approximations. Providence: AMS. ISBN 978-0-8218-4381-9.