Remark

Let $\nu$ be a left Haar measure on G and let Δ be its module. The mapping $f\mapsto f\cdot \nu$ is an isometric isomorphism of the algebra $L^1(G, \nu )$ onto A (loc. cit.). Thus one may also define Stell(G) as the enveloping stellar algebra of the involutive normed algebra $L^1(G, \nu )$.

Let G be a locally compact group and let $\nu$ be a left Haar measure on G. For $\mu\in \mathscr{M}^1(G)$ and $f\in L^2(G, \nu )$, one has $\mu*f\in L^2(G, \nu )$ (INT, VIII, §4, prop. 6). Let us then denote by $\boldsymbol{\gamma }(\mu)$ the endomorphism $f\mapsto \mu*f$ of $L^2(G, \nu )$. The mapping $\mu\mapsto \boldsymbol{\gamma }(\mu)$ is a representation of the algebra $\mathscr{M}^1(G)$ in the Banach algebra $\mathscr{L}(L^2(G, \nu ))$ of continuous endomorphisms of $L^2(G, \nu )$ (INT, VIII, §4, cor. to prop. 6). On the other hand, $\boldsymbol{\gamma }(\check{\mu})$ is the transpose of the endomorphism $\boldsymbol{\gamma }(\mu)$ (INT, VIII, §4, n$^o3$, prop. 8). It follows that $\boldsymbol{\gamma }(\mu^*)$ is the adjoint endomorphism of $\boldsymbol{\gamma }(\mu)$, and therefore that the mapping $\boldsymbol{\gamma }:\mu\mapsto \boldsymbol{\gamma }(\mu)$ is a morphism of involutive algebras of $\mathscr{M}^1(G)$ into the stellar algebra $\mathscr{L}(L^2(G, \nu ))$, called the left regular representation of $\mathscr{M}^1(G)$ in $L^2(G, \nu )$. According to INT, VIII, §4, n$^o7$, prop. 19, this representation is faithful.

Let $j$ be the canonical mapping of $L^1(G, \nu )$ into Stell(G). By restriction to $L^1(G, \nu )$, the regular representation $\boldsymbol{\gamma }$ defines an injective morphism of involutive algebras of $L^1(G, \nu )$ into $\mathscr{L}(L^2(G, \nu ))$, called the left regular representation of $L^1(G)$ in $L^2(G)$. According to prop. 20, there exists a unique morphism $\boldsymbol{\gamma }':$ Stell(G) $\rightarrow \mathscr{L}(L^2(G, \nu ))$ such that $\boldsymbol{\gamma }=\boldsymbol{\gamma }'\circ j$. One says that $\boldsymbol{\gamma }'$ is the left regular representation of Stell(G) in $L^2(G, \nu )$. By abuse of notation, we shall again write

for $f\in L^2(G, \nu )$ and $\varphi \in$ Stell(G). One has