Remark

The biregular representation of G in $L^p(G, \mu)$ is not necessarily faithful; its kernel is the image of the center of G under the mapping $g\mapsto (g, g)$ (exercise 4 of V, p. 487).

When $p= 2$, the left regular representation $\boldsymbol{\gamma }^{(2)}_G$ of G in the complex Hilbert space $L^2(G, \mu)$ is unitary, since it is isometric. Analogously, the right regular representation $\boldsymbol{\delta }^{(2)}_G$ in $L^2(G, \mu')$ is unitary.

We shall write simply $\boldsymbol{\gamma }_G=\boldsymbol{\gamma }^{(2)}_G$ and $\boldsymbol{\delta }_G=\boldsymbol{\delta }^{(2)}_G$, and we shall call these representations the left and right regular representations of G.

If G is unimodular, the biregular representation $\boldsymbol{\varrho }^{(2)}_G$ of $G\times G$ in $L^2(G, \mu)$ is unitary. We shall write it simply $\boldsymbol{\varrho }_G$.