Group C*-algebras of a discrete group

Let $\Gamma$ be a discrete group and $\mathbb C[\Gamma]$ its complex group algebra, with $\delta_g^*=\delta_{g^{-1}}$ . Its full group $C^*$ -algebra $C^*(\Gamma)$ is the completion in the supremum norm over all unitary representations of $\Gamma$ . Its reduced group $C^*$ -algebra $C_r^*(\Gamma)$ is the completion in the norm induced by the left regular representation on $\ell^2(\Gamma)$ . The phrase “group $C^*$ -algebra” is therefore ambiguous unless the full or reduced completion is specified.

Concrete construction

For a finitely supported function $f:\Gamma\to\mathbb C$ , write $f=\sum_g f(g)\delta_g$ . The left regular representation acts by

\lambda(f)\xi(s)=\sum_{g\in\Gamma}f(g)\xi(g^{-1}s).

Completing $\lambda(\mathbb C[\Gamma])$ in operator norm gives $C_r^*(\Gamma)$ . Taking the supremum over the integrated forms of all unitary representations gives $C^*(\Gamma)$ .

Comparison of the completions

Because the regular representation is among the representations used in the full norm, the identity on $\mathbb C[\Gamma]$ extends to a canonical surjective $*$ -homomorphism

C^*(\Gamma)\longrightarrow C_r^*(\Gamma).

It is an isomorphism exactly when $\Gamma$ is amenable. Hence the two constructions agree for finite, abelian, and more generally amenable discrete groups, but not for free groups on at least two generators.

Examples and conventions

For $\Gamma=\mathbb Z$ , Fourier transform identifies both completions with $C(\mathbb T)$ . For a finite group, the group $C^*$ -algebra is a finite-dimensional direct sum of matrix algebras determined by the irreducible representations. Both completions are unital for every discrete group, with unit $\delta_e$ . The notation $C^*(\Gamma)$ conventionally means the full completion, whereas $C_r^*(\Gamma)$ always means the reduced one.

References

Nathanial P. Brown and Narutaka Ozawa, $C^*$ -Algebras and Finite-Dimensional Approximations, American Mathematical Society, 2008. AMS DOI record. Relevant: §§2.5–2.6 on discrete group $C^*$ -algebras and amenability.
Kenneth R. Davidson, $C^*$ -Algebras by Example, American Mathematical Society, 1996. AMS DOI record. Relevant: examples of full and reduced group $C^*$ -algebras.