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,

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

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

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 $\mathcal{O}_2$.

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.