Theorem 1

Let I be a closed ideal of $L^1(G)$ distinct from $L^1(G)$. There exists a character $\widehat{x}\in \widehat{G}$ such that $\mathscr{F}_G(f)(\widehat{x}) = 0$ for every $f\in I$.

Comme the algebra $L^1(G)$ is regular (prop. 2 of II, p. 219) and without radical (cor. of prop. 22 of I, p. 126), and since the set of functions whose Fourier cotransform has compact support is dense in $L^1(G)$ (lemma 1), cor. 1 of I, p. 92 shows that the ideal I is contained in a regular maximal ideal of $L^1(G)$, that is to say in the kernel of a character $\widehat{y}$ of $L^1(G)$ (th. 2 of I, p. 30); one can then take $\widehat{x}=\widehat{y}^{-1}($cf. formula (8) of II, p. 207).

Corollary 1 (Wiener Tauberian Theorem)

Let $f\in L^1(G)$. If the Fourier transform of $f$ does not vanish, the functions $f*\varepsilon_x:g\mapsto f(gx^{-1})$, where $x$ runs through G, form a total set in $L^1(G)$ (EVT, I, p. 12, def. 1).

Let V be the closed vector subspace of $L^1(G)$ generated by the $f*\varepsilon_x$. By INT, VIII, §4, cor. of prop. 20, the space V is a closed ideal of $L^1(G)$. By th. 1 we have $V = L^1(G)$.