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Suppose $A$ is a commutative Banach algebra with identity and $I\subset A$ is an ideal. If there is a unique maximal ideal $M$ with $I\subset M$ does it follow that $I=M$? Or that $I$ is dense in $M$?

An answer assuming $A$ is semi-simple would be enough; in fact the case I'm really interested in is $A=L^1(G)$ where $G$ is a discrete abelian group. (So this seems related to Wiener's Tauberian Theorem; if I'm recalling the definition correctly WTT says that $\emptyset\subset\hat G$ is a set of spectral synthesis, while the current question asks whether singletons are sets of spectral synthesis.)

David C. Ullrich
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    Is it Wiener's theorem (as in the question) or Weiner's theorem (as in the name of the tag you created)? Moreover, is the theorem important enough to have its own tag? If a tag is needed, wouldn't one tag for Tauberian theorems be enough? – Martin Sleziak Aug 16 '16 at 08:07
  • @MartinSleziak Sorry about the spelling - I thought it was Wiener, then yesterday I saw it spelled Weiner somewhere online, was confused. Evidently I feel it deserves its own tag; it's a hugely fundamental thing in harmonic analysis. I'm not going to quarrel if It Is Determined that it shouldn't have its own tag. (It's not clear to me whether the criteria are such that there is such a thing as a theorem that's important enough to have its own tag; if there is such a thing I'd certainly put WTT in that class.) – David C. Ullrich Aug 16 '16 at 13:12
  • @MartinSleziak So I fixed the spelling. The misspelled tag should certainly be deleted - I don't know how to do that... – David C. Ullrich Aug 16 '16 at 13:14
  • There is nothing else that should be done about the misspelled tag. Tags with zero questions are removed by a script that runs once every 24 hours, see meta. – Martin Sleziak Aug 16 '16 at 13:53
  • One more thing - since you created the tag, it might be useful to create also tag-wiki or at least tag-excerpt. It might help other users to use the tag correctly. (This is probably not a problem here, since the tag name seems to be descriptive enough.) Another reasons is that the tags used on only one question are automatically deleted after certain time unless they have tag-wiki. – Martin Sleziak Aug 30 '16 at 05:09

1 Answers1

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As for your first question the answer is no, as there exist unital, commutative Banach algebras that are also local rings. For example, consider the unitisation of

$$R = \{f\in L_0((0,\infty))\colon \int\limits_0^\infty |f(t)|e^{-t^2}\, {\rm d}t <\infty\},$$ endowed with the convolution product:

$$(f\ast g)(t) = \int\limits_0^t f(t-s)g(s)\,{\rm d}s.$$

So $\{0\}$ is contained in $R$, the unique maximal ideal of $A$.

I will reply for the second part later.

Tomasz Kania
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