Unitary representations and representations of the full group C*-algebra

Statement

Let $G$ be a locally compact group. The integrated-form correspondence assigns to every strongly continuous unitary representation $U\colon G\to\mathcal U(H)$ the unique nondegenerate $C^*$ -representation of the full group $C^*$ -algebra

\widetilde U\colon C^*(G)\longrightarrow B(H)

extending $\widetilde U(f)=\int_G f(s)U_s\,ds$ for $f\in C_c(G)$ . Conversely, every nondegenerate representation of $C^*(G)$ arises uniquely this way. The two constructions preserve intertwining operators, direct sums, invariant subspaces, and unitary equivalence.

Recovering the group representation

The canonical map $s\mapsto u_s$ takes $G$ into the unitary multipliers of $C^*(G)$ . If $\pi$ is nondegenerate, its unique extension to the multiplier algebra gives $U_s=\overline\pi(u_s)$ . Nondegeneracy is essential here: it is what makes the multiplier extension and the recovered unitary representation canonical.

Full versus reduced

This correspondence is universal for the full group $C^*$ -algebra. The representation $\widetilde U$ factors through the canonical quotient onto $C_r^*(G)$ exactly when $U$ is weakly contained in the left regular representation. Therefore representations of the reduced algebra do not, in general, parameterize all unitary representations of $G$ .

References

D. P. Williams, Crossed Products of $C^*$ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: Sections 2.3–2.5 on integrated forms and group $C^*$ -algebras.
J. M. G. Fell and R. S. Doran, Representations of $*$ -Algebras, Locally Compact Groups, and Banach $*$ -Algebraic Bundles, vol. I, Academic Press, 1988. Publisher record. Relevant: the integration and disintegration of group representations.