Proposition 4

The Fourier transform and the Fourier cotransform are injective morphisms of involutive algebras of $L^1(G)$ into the algebra $\mathscr{C}_0(\widehat{G})$ of continuous functions vanishing at infinity on $\widehat{G}$.

The Fourier cotransform is a morphism of involutive algebras of $L^1(G)$ into the algebra of bounded continuous functions on $\widehat{G}$ (prop. 3). Since it identifies with the Gelfand transform, its image is contained in $\mathscr{C}_0(\widehat{G})$ (I, p. 37, prop. 5), and its kernel is the radical of $L^1(G)$ (prop. 8 of I, p. 38), which is zero (cor. of prop. 22 of I, p. 126).

We shall see later (corollary to proposition 13 of II, p. 221) that the cotransform of Fourier on $\mathscr{M}^1(G)$ is also injective.