Proposition 11

— Let $G$ be a locally compact group, $E$ a quasi-complete locally convex space, $U$ a continuous representation of $G$ in $E$. (i) If $\lambda \in \mathscr{C}'(G)$, $\mu \in \mathscr{C}'(G)$, then $U(\lambda * \mu) = U(\lambda)U(\mu)$. (ii) Suppose that $E$ is a Banach space, and let $\rho(s) = \|U(s)\|$ for $s \in G$. If $\lambda \in \mathscr{M}^\rho(G)$, $\mu \in \mathscr{M}^\rho(G)$, then $U(\lambda * \mu) = U(\lambda)U(\mu)$. Let $\lambda, \mu$ be in $\mathscr{C}'(G)$. For any $x \in E$ one has, by applying notably Props. 1 and 4 of Ch. VI, §1, No. 1,

whence (i). An analogous argument may be applied in case (ii).

With G still a locally compact group, let us assume that G operates continuously on the left in a locally compact space X. This defines (§2, No. 4) a continuous linear representation $\gamma$ of G in $\mathscr{M}(X)$ (equipped with the topology of compact convergence in $\mathscr{H}(X)$).