Proposition 16

— Let $U$ be a continuous representation of $G$ in a Banach space $E$. Set $\rho(s) = \|U(s)\|$ for all $s \in G$. For every $f \in L^\rho(G, \beta)$, set $U(f) = U(f \cdot \beta)$. Then $f \mapsto U(f)$ is a linear representation of the algebra $L^\rho(G, \beta)$ in $E$, such that $\|U(f)\| \leq \|f\|_\rho$.

This follows from §2, No. 6 and §3, No. 3, Prop. 11.