Proposition 3

— Assume $\chi$ continuous. Let $\mu \in \mathscr{C}'(G)$, $f \in \mathscr{C}(X)$. Then: (i) $\mu$ and $f$ are convolvable relative to $\beta$. (ii) Formula (3) of No. 1 defines for every $x \in X$ a convolution product $\mu *_{\beta} f$ that is continuous and is none other than the element $\gamma_\chi(\mu)f$ defined by the continuous representation $\gamma_\chi$ of $G$ in $\mathscr{C}(X)$; moreover, the mapping $(\mu, f) \mapsto \mu *_{\beta} f$ is hypocontinuous relative to the equicontinuous subsets of $\mathscr{C}'(G)$ and the compact subsets of $\mathscr{C}(X)$. (iii) If in addition $f \in \mathscr{K}(X)$, then the product $\mu *_{\beta} f$ of (ii) belongs to $\mathscr{K}(X)$ and the mapping $(\mu, f) \mapsto \mu *_{\beta} f$ is hypocontinuous relative to the equicontinuous subsets of $\mathscr{C}'(G)$ and the compact subsets of $\mathscr{K}(X)$.

We know that $\mu$ and $f$ are convolvable (§3, No. 2, Prop. 8 (i)). On the other hand, with the notations of §2, we have

since $\mathscr{C}(X)$ is complete. In particular, for every $x \in X$, This, combined with Prop. 2 (i), and §2, No. 6, proves (ii). Finally, if $f \in \mathscr{K}(X)$ then $\mu * (f \cdot \beta)$ has compact support (§3, No. 2, Prop. 9), therefore $\mu *_{\beta} f \in \mathscr{K}(X)$. For, let us consider the continuous representation $U$ of $G$ in the completion $\mathscr{K}(X)^{\sim}$ obtained by extending by continuity the continuous operators $\gamma_\chi(s)$ in $\mathscr{K}(X)$ (§ 2, No. 1, Remark 3). Let $S$ be the support of $\mu$. The functions $\gamma_\chi(s)f$, for $s \in S$, have their support contained in a fixed compact set $K$. The set $\mathscr{K}(X, K)$ is a complete linear subspace of $\mathscr{K}(X)$. Therefore $U(\mu)f \in \mathscr{K}(X)$. One sees as before that $U(\mu)f = \mu *^\beta f$, and (iii) again follows from § 2, No. 6.