Nuclear C*-algebras are exact - Imported Knowlpedia Content

Statement

Every nuclear $C^*$ -algebra is an exact $C^*$ -algebra. Explicitly, if $A$ is nuclear and

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

is a [[algebra-modules/short-exact-sequence|short exact sequence of $C^*$ -algebras]], then

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

is exact. Nuclearity is strictly stronger: exactness alone neither forces the minimal and maximal tensor products with $A$ to agree nor supplies completely positive finite-dimensional approximations Brown–Ozawa, Proposition 2.3.8.

Proof idea

Nuclearity identifies the minimal tensor product with a tensor product whose quotient behavior is controlled by completely positive approximations. Equivalently, approximate the identity map on $A$ pointwise in norm by completely positive contractions through matrix algebras. Tensoring those factorizations with the given extension reduces the kernel calculation to matrix amplifications, which preserve exactness. Passing to the point-norm limit proves that the kernel in $B\otimes_{\min}A$ is exactly $I\otimes_{\min}A$ .

Strictness of the implication

Commutative $C^*$ -algebras and compact-operator algebras are nuclear, hence exact. The converse fails: $C_r^*(\mathbb F_2)$ , the reduced $C^*$ -algebra of the free group on two generators, is exact but not nuclear. In the reduced group setting, nuclearity detects amenability, whereas exactness holds for a broader class of groups Brown–Ozawa, §§2.3 and 5.1.

Logical role

This result is an implication between properties, not an alternative definition of nuclearity. In arguments about extensions, exactness is often the precise hypothesis needed; assuming nuclearity may introduce unnecessary structure.

References

Nathanial P. Brown and Narutaka Ozawa, C-Algebras and Finite-Dimensional Approximations*, American Mathematical Society, 2008. AMS DOI record. Relevant: §2.3, especially Proposition 2.3.8, on nuclearity and exactness.