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I would like to know why we have the equivalence between the following three statements of the Wiener-Tauberian theorem:

version 1: If $I$ is a closed ideal in $L^1(\mathbb R)$, such that the set $hull(I)=\{\varphi: L^1(\mathbb R)\to \mathbb C \, \mbox{homomorphism}: \varphi(x) = 0\, \forall x\in I\}=\emptyset \, $, then: $I=L^1(\mathbb R)$.

version 2: Every proper closed ideal in $L^1(\mathbb R)$ is contained in a maximal ideal.

version 3: If $f\in L^1(\mathbb R)$, such that its Fourier transform $\hat f$ is a nowhere vanishing function on $\mathbb R$, then the closed ideal generated by f is whole of $L^1(\mathbb R)$, i.e., $\overline{span}\{f_{\tau},\, \tau \in \mathbb R\}= L^1(\mathbb R)$.

Thank you in advance

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    $\begingroup$ can you post the sources for each of these three statements? $\endgroup$
    – john mangual
    Commented Aug 3, 2017 at 21:22
  • $\begingroup$ @John, You can find them in every article that deals with the Wiener-Tauberian theorem $\endgroup$
    – Z. Alfata
    Commented Aug 4, 2017 at 15:49

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