Proposition 5

By continuity, the Fourier transform and the cotransform of Fourier extend uniquely to isomorphisms of stellar algebras of Stell(G) onto $\mathscr{C}_0(\widehat{G})$.

The cotransform of Fourier extends by continuity to a morphism of stellar algebras from Stell(G) into $\mathscr{C}_0(\widehat{G})$. If one identifies $\widehat{G}$ with $\mathsf{X}$(Stell(G)) (cor. 1 of II, p. 204 and prop. 1 of II, p. 202), this extension is the Gelfand transform of Stell(G). By th. 1 of I, p. 108, it is an isomorphism. The assertion concerning the Fourier transform follows from this.

We shall always denote by $\overline{\mathscr{F}}$ and $\mathscr{F}$ the isomorphisms of prop. 5.