Full group C*-algebra

Let $G$ be a locally compact group. For $f$ in the $L^1$ group algebra, define

\|f\|_{\max}=\sup_U\left\|\int_G f(s)U_s\,ds\right\|,

where $U$ ranges over all strongly continuous unitary representations of $G$ . The full group $C^*$ -algebra $C^*(G)$ is the completion of $L^1(G)/\ker\|\cdot\|_{\max}$ in this norm. Equivalently, it is the enveloping $C^*$ -algebra obtained from every integrated form, rather than from one selected representation.

Universal representation property

Every strongly continuous unitary representation $U$ of $G$ integrates uniquely to a nondegenerate representation of the $C^*$ -algebra $C^*(G)$ . Conversely, every nondegenerate representation of $C^*(G)$ comes from a unique $U$ . For nondiscrete $G$ , the individual operators $U_s$ are generally represented by unitaries in the multiplier algebra, not by elements of $C^*(G)$ itself.

Full versus reduced

The left regular representation supplies a canonical quotient

C^*(G)\twoheadrightarrow C_r^*(G)

onto the reduced group $C^*$ -algebra. This quotient is an isomorphism when $G$ is amenable; it need not be injective for a nonamenable group. The adjective “full” records that all unitary representations contribute to the defining norm.

Standard examples

For a discrete group, $C^*(G)$ is the universal $C^*$ -algebra generated by unitaries satisfying the group relations. For a locally compact abelian group, Fourier transform identifies $C^*(G)$ with $C_0(\widehat G)$ , where $\widehat G$ is the Pontryagin dual.

References

Dana P. Williams, Crossed Products of $C^*$ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: chapters 1–2 and appendix A on group algebras, integrated forms, and nondegenerate representations.
Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd edition, CRC Press, 2016. DOI record. Relevant: chapters on unitary representations and group $C^*$ -algebras.