Let $(\pi, V)$ be a representation of the group $G$. To make the setting as general as possible, I will not put any restrictions on $\pi, V$, and $G$ from the beginning.
By the very definition, for each $g\in G$, $\pi(g)\in \operatorname{GL}(V)$. I would like to use $\pi$ to construct all maps in $\operatorname{End}(V)$ (or rather $B(V)$ the set of bounded functions on $V$).
Clearly, $\pi(\phi)=\int_G\phi(g)\pi(g)\, dg$ is one such function in $\operatorname{End}(V)$ when $\phi\in L^1(G)$. However, in this generality we cannot necessarily generate all endomorphisms; for instance when $\pi=I_V$.
My question is
Under what conditions the equality $B(V)=\{\pi (\phi)\}$ holds?