Proposition 11

a) Let $u= (u_{\pi})_{\pi\in\widehat{G}}$ be an element of $F(\widehat{G})$. If the family $(\mathscr{F}_{\pi}(u_{\pi}))_{\pi\in\widehat{G}}$ is uniformly summable in $\mathscr{C}(G)$, then its sum $f$ is a continuous function on G whose Fourier cotransform is $u$;

b) Let $f\in L^1(G)$. If the family $(\mathscr{F}_{\pi}(\pi (f)))_{\pi\in G}$ is uniformly summable in $\mathscr{C}(G)$, then its sum is a continuous fun$\widehat{c}$tion on G whose class in $L^1(G)$ is equal to $f$.

Let us prove assertion a). Since G is compact, the sum $f$ of the family $(\mathscr{F}_{\pi}(u_{\pi}))$ belongs to $L^2(G)$ and the series

converges in $L^2(G)$ to $f$ (INT, IV, p. 127, § 3, No.$^o3$, prop. 4). Hence one has $\mathscr{F}_G((u_{\pi})_{\pi}) =f$ in $L^2$(G), whence $(u_{\pi})_{\pi\in\widehat{G}}=\overline{\mathscr{F}}_G(f)$ (Theorem 2).

Let us prove assertion b). Assertion a) may be applied to the family $(\pi (f))_{\pi\in G}$. The sum $g$ of the family $(\mathscr{F}_{\pi}(\pi (f)))_{\pi\in G}$ is a continuous function, hence belonging to $L^2$(G), such that $\mathscr{F}_G(g) = (\pi (f))_{\pi\in G}$. Since $\mathscr{F}_G$ is injective on $L^2$(G), one has $f=g$ in $L^2$(G), hence in $L^1(G)$.

The algebra $\mathscr{C}(\widehat{G})$ of complex-valued functions on $\widehat{G}$ is identified with the center of the algebra $F(\widehat{G})$ by the mapping which to $f:\widehat{G}\rightarrow \mathbf{C}$ associates the central element $(f(\pi )1_{E_{\pi}})_{\pi\in\widehat{G}}$ of $F(\widehat{G})$. Let us denote by $\mathscr{M}^1(G^{\sharp})$ the space of bounded central measures on G. It is a closed subspace of the Banach algebra $\mathscr{M}^1(G)$. For every measure $\nu \in \mathscr{M}^1(G^{\sharp})$ and every representation $\pi \in \widehat{G}$, one has $\pi (\nu )\in$ End$_G(E_{\pi})$ hence $\pi (\nu )$ is a multiple of the identity mapping of $E_{\pi}$ by Schur’s Lemma (Prop. 6 of V, p. 386). The restriction of the Fourier cotransform to $\mathscr{M}^1(G^{\sharp})$ is therefore identified with a unital morphism of involutive Banach algebras from $\mathscr{M}^1(G^{\sharp})$ into $\mathscr{C}(\widehat{G})$. This morphism is injective; the image of $L^1(G^{\sharp})$ is contained in $\mathscr{C}_b(\widehat{G})$ and its restriction to $L^2(G^{\sharp})$ is an isometric isomorphism onto $L^2(\widehat{G},\widehat{\mu})$.