Proposition 5

Suppose that each function $\chi(s, \cdot)$ is bounded. Then $\gamma_\chi$ leaves $\overline{K(X)}$ stable, and the linear representation $\gamma_\chi$ of G in $\overline{K(X)}$ is continuous.

It is clear that $\gamma_\chi$ leaves $\overline{K(X)}$ stable and that each of the $\gamma_\chi(s)$ is continuous in $\overline{K(X)}$. On the other hand, for every $f \in K(X)$, $s \mapsto \gamma_\chi(s)f$ is a continuous mapping of G into $K(X)$ and a fortiori into $\overline{K(X)}$. Therefore the representation $\gamma_\chi$ in $\overline{K(X)}$ is continuous (No. 1, Prop. 2).