Nuclearity of reduced group C*-algebras and amenability for discrete groups

Statement

Let $\Gamma$ be a discrete group. The Lance theorem states that

\Gamma\text{ is amenable} \quad\Longleftrightarrow\quad C_r^*(\Gamma)\text{ is nuclear},

where amenability is in the sense of an invariant mean and $C_r^*(\Gamma)$ is the reduced group $C^*$ -algebra generated by the left regular representation on $\ell^2(\Gamma)$ . Thus amenability of a discrete group is equivalent to an intrinsic finite-dimensional approximation property of its reduced operator algebra.

Meaning of nuclearity

For a $C^*$ -algebra $A$ , nuclearity means that for every $C^*$ -algebra $B$ , the maximal and minimal $C^*$ -tensor norms on the algebraic tensor product $A\odot B$ agree. Equivalently, the identity on $A$ is a point-norm limit of completely positive contractions factoring through matrix algebras. Applied to $C_r^*(\Gamma)$ , this analytic property detects the existence of invariant averaging on $\Gamma$ .

Proof directions and scope

Følner-type approximations for an amenable group yield finite-rank completely positive approximations of the regular representation, giving nuclearity. The converse extracts invariant averaging from nuclear approximation of the reduced algebra; this is the direction associated with Lance's theorem. The word “discrete” is part of this formulation and should not be silently discarded when quoting the result.

Consequences

For the free group $F_2$ , nonamenability implies that $C_r^*(F_2)$ is not nuclear. For an amenable discrete group, the canonical quotient from the full to the reduced group $C^*$ -algebra is an isomorphism and the common algebra is nuclear. Equality of the full and reduced completions and nuclearity are related conclusions here, but they are distinct properties for general $C^*$ -algebras.

References

E. Christopher Lance, “On nuclear $C^*$ -algebras,” Journal of Functional Analysis 12 (1973), 157–176. DOI record. Relevant: the nuclearity criterion for reduced $C^*$ -algebras of discrete groups.
Nathanial P. Brown and Narutaka Ozawa, $C^*$ -Algebras and Finite-Dimensional Approximations, American Mathematical Society, 2008. AMS DOI record. Relevant: §2.6, especially Theorem 2.6.8.