Talk:Banach algebra

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I cut "The algebra is unital if and only if the original space is compact. Also, since every continuous function on a compact space is automatically bounded, we do not need to assume the boundedness of the functions in this case." from the example "bounded continuous functions because the first sentance isn't true. The function identically 1 will be bounded and continuous on any space and act as a unit to this algebra (which is C(X) for X the stone-cech compactification of our original space). I think whoever added it was thinking of the continuous functions vanishing at infinity, for which the above would hold. A Geek Tragedy 16:50, 25 August 2006 (UTC)

What is a regular Banach algebra? A normal Banach algebra? Links to answers should possibly be added to the pages "normal" and "regular" too. --Tilin 11:59, 19 October 2006 (UTC)

Would a section on Banach *-algebras be worth including? This would have the side-effect of killing off the current misleading entry on B*-algebras which is to my knowledge non-standard. NowhereDense (talk) 10:33, 27 September 2008 (UTC)

Are you saying that "B*-algebra" is not a standard term? My understanding is that the terminology is historical; it was studied heavily at one point, but after we found its representation to, I forgot what, it hasn't been studied much. Anyway to answer your question, I don't there is any problem discussing a *-algebras that is a Banach algebra. -- Taku (talk) 13:26, 27 September 2008 (UTC)

My understanding is that "B*-algebra" historically meant what is now called "abstract C*-algebra" -- however, the history is a little complicated because the original axioms were slightly different from what they are now, and the removal of redundant axioms is I think due to work of Arens and possibly others. My problem is that the current wikipedia entry on B*-algebra is not using the term in what I believe to be the correct way. I will try to update it with a citation to back up these claims... NowhereDense (talk) 20:48, 28 September 2008 (UTC)

Well, that's not how I understand B*-algebras :) I thought B*-algebras and C*-algebras are, historically, studied separately because of the difference in the axioms; every C*-algebra is B*, but the converse is not true. But then it turned out that B* isn't much different from C*, and so studying C* is sufficient to understand B*. So, the definition given in the B*-algebra is exactly one I learned (or I thought I learned). In any case, what we need are references; this discussion gets us nowhere. -- Taku (talk) 12:10, 30 September 2008 (UTC)

You are quite right, in that what we need are some references for the history on this subject -- unfortunately I have limited access to the relevant texts for the next few weeks, volume 1 of Dunford and Schwarz would seem an obvious place to look. By the way, when you say the defn in the B*-algebra entry is the one you remember, do you mean the one dated 28th September -- because that one is my edit ;-) -- or do you mean the earlier one? —Preceding unsigned comment added by NowhereDense (talk • contribs) 22:45, 30 September 2008 (UTC)

Ideals and ....

I think, restricting to the unital case is a pretty bad idea here. Anyway, there's a much beeter and more general description of the gelfand represenation at Gelfand representation -- Roman3 (talk) 14:36, 11 June 2010 (UTC)

Quasi-Banach algebra

In the F-space article, the term quasi-Banach algebra is mentioned. What is that? Perhaps someone could include the definition in the article? Isheden (talk) 07:22, 2 September 2011 (UTC)

My pure speculation (i.e., no source) is that you require $\|xy\|\leq c\|x\|\|y\|$. If $c=1$, then it is a Banach algebra. In any rate, this should be mentioned in the article. -- Taku (talk) 16:36, 2 September 2011 (UTC)

I found this: [1] - possibly the definition fits better in the quasinorm article? Isheden (talk) 20:02, 2 September 2011 (UTC)

Maybe. At least I don't think standard texts (e.g., Rudin) discusses this type of algebra. -- Taku (talk) 00:04, 5 September 2011 (UTC)

I have added the definition in the quasinorm article now. Isheden (talk) 07:53, 5 September 2011 (UTC)

Examples

the space of (complex-valued) continuous functions on a locally compact (Hausdorff) space that vanish at infinity. It took me a while to figure out that it is the functions, rather than the space, that vanish at infinity. Can the text be clarified? Also, what is the X in C₀(X)? Bo Jacoby (talk) 09:44, 3 January 2013 (UTC).

10 years late, but I clarified this. I also changed locally compact (Hausdorff) space to locally compact Hausdorff space for clarity. --Caliburn · (Talk · Contribs · CentralAuth · Log) 14:01, 7 September 2023 (UTC)

Nonarchimedean case?

It would be nice to mention a definition for nonarchimedean fields as used in rigid analytic spaces Math MisterY (talk) 02:10, 9 April 2015 (UTC)