Exact C* algebra - Alchetron, The Free Social Encyclopedia

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

Definition

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

0 → A → f B → g C → 0

the sequence

0 → A ⊗ min E → f ⊗ id B ⊗ min E → g ⊗ id C ⊗ min E → 0 ,

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

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 separable C*-algebra A is exact if and only if it is isomorphic to a subalgebra of the Cuntz algebra O 2 .

All nuclear C*-algebras and their C*-subalgebras are exact.

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