Exact C*-algebra

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,

${\displaystyle 0\;{\xrightarrow {}}\;A\;{\xrightarrow {f}}\;B\;{\xrightarrow {g}}\;C\;{\xrightarrow {}}\;0}$

the sequence

${\displaystyle 0\;{\xrightarrow {}}\;A\otimes _{\min }E\;{\xrightarrow {f\otimes \operatorname {id} }}\;B\otimes _{\min }E\;{\xrightarrow {g\otimes \operatorname {id} }}\;C\otimes _{\min }E\;{\xrightarrow {}}\;0,}$

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 ${\displaystyle {\mathcal {O}}_{2}}$.

References

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