This article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects:

Articles for creation This article was reviewed by member(s) of WikiProject Articles for creation. The project works to allow users to contribute quality articles and media files to the encyclopedia and track their progress as they are developed. To participate, please visit the project page for more information.Articles for creationWikipedia:WikiProject Articles for creationTemplate:WikiProject Articles for creationAfC This article was accepted on 2 June 2010 by reviewer Chevymontecarlo (talk · contribs). Mathematics Low‑priority This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Low This article has been rated as Low-priority on the project's priority scale.

As far as I understand, Wiener's 1/f theorem and Wiener's tauberian theorem are the same mathematical statement. Perhaps these should be merged? Sasha (talk) 06:08, 28 June 2011 (UTC)[reply]

That should just be Gelfand's proof; there's no "different" proof. The result follows from the explicit identification of the characters, not the other way around as stated. The general fact used here is that the Gelfand map preserves the spectrum. Also, there's just Banach algebras, no C*-algebras; this is what makes the identification of characters nontrivial. (Completion of A(T) with respect to the sup norm is the C*-algebra C(T) but this is not relevant here.) Mct mht (talk) 10:04, 14 August 2012 (UTC)[reply]

could you please elaborate? To which paragraph are you referring?

Wiener's 1/f theorem has plenty of different proofs (starting from that of Wiener); most of them do not use the theory of Banach algebras at all.

Sasha (talk) 18:12, 14 August 2012 (UTC)[reply]

I was refering to the short last section. There should just be one proof, due to Gelfand, that uses Banach algebra, not C*-algebra, techniques, and a key step in the proof is identification of the space or characters, or maximal ideals (this is nontrivial precisely because A(T) is not a C*-algebra). So what's being stated reads backwards and not quite accurate.

Also, in defining the Banach algebra structure on A(T), it would be probably better if the article starts by pointing out that the Fourier-Gelfand transform is injective on l^1(Z) and A(T) is nothing but its image. It would follow trivially then, for example, A(T) is closed under multiplication, A(T) sits in C(T) and the inclusion is a norm decreasing map (because the l^1 norm dominates the C*-norm), etc. Again, the article right now has it backwards. Mct mht (talk) 08:06, 15 August 2012 (UTC)[reply]

I have started from your first suggestion. Could you please have a look at the last section now and say whether the new wording is a step forward or backward? Thanks, Sasha (talk) 01:42, 18 August 2012 (UTC)[reply]

Forward. :) Mct mht (talk) 06:34, 18 August 2012 (UTC)[reply]