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 &otimes;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 $$\mathcal{O}_2$$.