Exact C*-algebra - Imported Knowlpedia Content

Definition

A $C^*$ -algebra $A$ is exact if tensoring with $A$ by the minimal $C^*$ -tensor product preserves every short exact sequence: whenever

0\longrightarrow I\longrightarrow B\longrightarrow B/I\longrightarrow 0

is exact, so is

0\longrightarrow I\otimes_{\min}A\longrightarrow B\otimes_{\min}A\longrightarrow (B/I)\otimes_{\min}A \longrightarrow 0.

Thus the first map must remain injective and its image must be exactly the kernel of the quotient map. The last map is always surjective; exactness is the assertion that minimal tensoring introduces no additional kernel.

Equivalent formulations

It is enough to test canonical ideal–quotient sequences. For every closed two-sided ideal $I\triangleleft B$ , there is a canonical surjection

(B\otimes_{\min}A)/(I\otimes_{\min}A) \longrightarrow (B/I)\otimes_{\min}A.

The algebra $A$ is exact precisely when this map is injective, hence a $*$ -isomorphism, for every pair $(B,I)$ . Exactness can equivalently be placed in the first tensor factor because the minimal tensor product is symmetric Brown–Ozawa, §2.3.

Permanence and examples

Exactness passes to $C^*$ -subalgebras, quotients, and inductive limits, although it is not a general three-space property for extensions. Every nuclear $C^*$ -algebra is exact, but the converse fails. For example, the reduced group $C^*$ -algebra $C_r^*(\mathbb F_2)$ of the free group on two generators is exact and nonnuclear. This example separates exactness, which concerns the behavior of the minimal tensor product on extensions, from nuclearity, which requires agreement of the minimal and maximal tensor norms in every tensor product Brown–Ozawa, Chapter 2.

Conventions and scope

References

Nathanial P. Brown and Narutaka Ozawa, C-Algebras and Finite-Dimensional Approximations*, American Mathematical Society, 2008. AMS DOI record. Relevant: §2.3 on exact $C^*$ -algebras and their permanence properties.
Eberhard Kirchberg, “The Fubini theorem for exact C-algebras,” Journal of Operator Theory* 10 (1983), 3–8. Journal record. Relevant: the tensor-product characterization of exactness.