Definition 1

For every measure $\nu \in \mathscr{M}^1(G)$, the element $(\pi (\nu ))_{\pi\in\widehat{G}}$ of $F_b(\widehat{G})$ is denoted by $\mathscr{F}_G(\nu )$. The map of $\mathscr{M}^1(G)$ into $F_b(\widehat{G})$ thus defined is called the Fourier cotransform of G, and $\overline{\mathscr{F}}_G(\nu )$ is called the Fourier cotransform of the measure $\nu$.

For $f\in L^1$(G), we shall write $\overline{\mathscr{F}}_G(f) =\overline{\mathscr{F}}_G(f\cdot \mu)$.

For every representation $\pi \in \widehat{G}$, the map $\nu \mapsto \pi (\nu )$ is a unital morphism of involutive Banach algebras of $\mathscr{M}^1(G)$ into End(E$_{\pi}$) (Lemma 1 of V, p. 401). The Fourier cotransform is therefore a unital morphism of involutive Banach algebras of $\mathscr{M}^1(G)$ into $F_b(\widehat{G})$.