Canonical quotient from full to reduced group C*-algebra

Definition

Let $G$ be a locally compact group and let $\lambda$ be its left regular representation. Since the universal norm on $C_c(G)$ dominates the norm $f\mapsto\|\lambda(f)\|$ , the assignment $f\mapsto\lambda(f)$ extends uniquely from the full group $C^*$ -algebra to the reduced group $C^*$ -algebra as a surjective $*$ -homomorphism

q_G\colon C^*(G)\longrightarrow C_r^*(G).

This map is the canonical full-to-reduced quotient. Its kernel is precisely the kernel of the integrated regular representation, so $C_r^*(G)\cong C^*(G)/\ker\lambda$ .

Representation-theoretic meaning

A nondegenerate representation of $C^*(G)$ factors through $q_G$ exactly when the corresponding unitary representation of $G$ is weakly contained in $\lambda$ . Thus the reduced algebra records only those representations detected by the regular representation, whereas the full algebra records every strongly continuous unitary representation.

When the quotient is an isomorphism

The map $q_G$ is an isomorphism exactly when $G$ is amenable. Without that hypothesis, its kernel can be nonzero; consequently, the notation “the group $C^*$ -algebra” should not be used to identify the full and reduced completions. For example, nonabelian free groups give standard cases in which the quotient is not injective Brown–Ozawa, Chapter 2.

References

D. P. Williams, Crossed Products of $C^*$ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: Section 2.5 on full and reduced group $C^*$ -algebras.
N. P. Brown and N. Ozawa, $C^*$ -Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. DOI record. Relevant: Chapter 2 on reduced group $C^*$ -algebras and amenability.