Remark

In general, the left regular representation $\boldsymbol{\gamma }'$ of Stell(G) in $L^2(G, \nu )$ is not faithful. One can prove that it is so if and only if there exists on $L^{\infty}_{\mathbf{R}}(G)$ a positive linear form $f$ such that $f(1) = 1$ and $f(\boldsymbol{\gamma }(g)x) =f(x)$ for every $(g, x)\in G\times G$ (one then says that the group G is amenable, cf. EVT, IV, p. 73, exercise 4).