Let $G$ be a locally compact group. Suppose $1<p<\infty$. We denote by $CV_p(G)$ the space of operators $T$ on $L^p(G)$ such that $T(f*g)=(Tf)*g$.
1) In the book "Amenable locally compact groups" of Pier, Proposition 9.2 (page 83), the author asserts that each operator of $CV_p(G)$ is a convolution operator with a measure on $G$. This proposition seems false to me. Does is it true?
2) (If the proposition 9.2 of Pier is false) Suppose that $G$ is a discrete group. Let $T\in CV_p(G)$. Does there exist a measure $\mu$ on $G$ such that $T(g)=\mu*g$ for any continuous function $g$ with compact support?
3) Let $G$ be an abelian locally compact group. We denote $\varepsilon_a$ the Dirac measure in the point $a\in G$. If $|\lambda|=1$, the operator $\lambda(\varepsilon_a*\cdot)$ is a extremal point of the closed unit ball of $CV_p(G)$.
Does there exist other known exemples of extremal points of this ball? Does there exist references on this subject?