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,

$0\stackrel{}{\to }A\stackrel{f}{\to }B\stackrel{g}{\to }C\stackrel{}{\to }0$

the sequence

$0\stackrel{}{\to }A{\otimes }_{\min }E\stackrel{f\otimes \mathrm{id}}{\to }B{\otimes }_{\min }E\stackrel{g\otimes \mathrm{id}}{\to }C{\otimes }_{\min }E\stackrel{}{\to }0,$

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

Properties

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.

References

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