Lemma 4

Let $p$ be a real number $\geqslant 1$. For $f\in \mathscr{L}^1(G)$, the linear mapping $\boldsymbol{\gamma }^{(p)}_G(f)$ coincides with the endomorphism of $L^p(G)$ defined by $\varphi \mapsto f*^\mu\varphi$.

This follows from INT, VIII, p. 157, § 4, No.$^o2$, Prop. 6, taking account of formula (14) of INT, VIII, p. 165.