Proposition 8

The linear representation $\gamma_{\chi,p}$ of $G$ in $L_C^p(X, \beta)$ is continuous and isometric.

Let $f \in \mathscr{K}(X)$. When $s$ tends to $s_0$ in $G$, $\gamma_{\chi,p}(s)f$ tends to $\gamma_{\chi,p}(s_0)f$ in $\mathscr{K}(X)$, hence in $L_C^p(X, \beta)$. Since the $\gamma_{\chi,p}(s)$ are isometric, Prop. 8 is obtained by applying Remark 2 of No. 1.

For the case that $\chi$ is not assumed continuous, cf. §4, Exer. 13.