Exact C*-algebra

Definition

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

:0 ;\xrightarrow{}; A ;\xrightarrow{f}; B ;\xrightarrow{g}; C ;\xrightarrow{}; 0

the sequence

:0;\xrightarrow{}; A \otimes_\min E;\xrightarrow{f\otimes \operatorname{id}}; B\otimes_\min E ;\xrightarrow{g\otimes \operatorname{id}}; C\otimes_\min E ;\xrightarrow{}; 0,

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

Properties

• Every nuclear C*-algebra is exact.

• 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.

• It follows that every sub-C*-algebra of a nuclear C*-algebra is exact.

Characterizations

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 \mathcal{O}_2.

References

• {{cite book