Lemma 3

Let $p$ be a real number $\geqslant 1$. The left (resp. right) regular representation of G in $L^p(G, \mu) ($resp. in $L^p(G, \mu'))$ is faithful.

More precisely, let $q$ be the conjugate exponent of $p$ and let $g$ be an element of G such that $g\not =e$.

a) There exists a function $\varphi \in \mathscr{K}(G)$, positive and nonzero in $L^q(G, \mu)$, such that $\langle \varphi ,\boldsymbol{\gamma }^{(p)}_G(g)\varphi \rangle = 0$;

b) There exists a function $\varphi \in \mathscr{K}(G)$, positive and nonzero in $L^q(G, \mu')$, such that $\langle \varphi ,\boldsymbol{\delta }^{(p)}_G(g)\varphi \rangle = 0$.

Let us consider the case of the left regular representation $\boldsymbol{\gamma }^{(p)}_G$, that of the right regular representation being similar. Assertion a) implies that $\boldsymbol{\gamma }^{(p)}_G$ is faithful, for if $g\not =e$ is an element of G, and if $\varphi$ is as in a), one has $\varphi \not =\boldsymbol{\gamma }^{(p)}_G(g)\varphi$ since $\langle \varphi , \varphi \rangle =\int_G\varphi^2>0$.

Let us therefore prove a). Let $g\not =e$ in G. Let C be a symmetric compact neighbourhood of $e$ such that $g \notin C^2$ and let $\varphi \in \mathscr{K}(G)$ be a positive continuous function of integral 1 with support contained in C; the function $\varphi$ is nonzero in $L^p(G, \mu)$. Since $C\cap g^{-1}C =\emptyset$, one has

Let $\varrho$ be a continuous bounded representation of G in a Banach space E. For every $f\in L^1(G)$ (resp. $f'\in L^1(G, \mu')$) and every $g\in G$, one has

(1) $\varrho (g)\varrho (f\cdot \mu) =\varrho (\varepsilon_g*(f\cdot \mu)) =\varrho (\boldsymbol{\gamma }^{(1)}_G(g)f\cdot \mu)$,

(2) $\varrho (f'\cdot \mu')\varrho (g) =\varrho ((f'\cdot \mu')*\varepsilon_g) =\varrho (\boldsymbol{\delta }^{(1)}_G(g^{-1})f'\cdot \mu')$

(INT, VIII, p. 144, § 3, n$^o2$, formula (5)).

Suppose G unimodular, and put $\mu'=\mu$. The biregular representation of G in $\mathscr{L}^p(G)$ (resp. in $L^p(G)$) is the representation $\boldsymbol{\varrho }^{(p)}_G$ of $G\times G$ in $\mathscr{L}^p(G)$ (resp. in $L^p(G)$) such that

It is a continuous linear representation (Lemma 1 of V, p. 377). It satisfies

for every $f\in \mathscr{L}^p$(G), every $(g_1, g_2)\in G\times G$ and every $x\in G$.