# Should there be a “Banach algebras” tag?

Apologies if this is isomorphic to an existing question.

I ask because it seems like there have been some recent questions that clearly belong better under the tag "Banach algebras" than "Banach spaces", yet where adding an "operator algebras" tag seems to stretch the point a bit far. A quick search for "Banach algebra" on the main site throws up about 40 questions containing the phrase, many of which might benefit from the tag being used.

Is there any reason why there isn't a banach-algebras tag?

Here are some candidates where I think the tag is both appropriate and useful:

Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

a question about invertibility of Banach Algebra

How far is a Banach algebra from being a multiplicative group?

Why is $GL(B)$ a Banach Lie Group?

Closure of the invertible operators on a Banach space

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\}$

Update: Since there was no objection to the creation of the tag and I can't think of a single reason not to have it, I now took the initiative and created and I will slowly start adding it to questions that should have it. If anyone else sees a question that should have the tag please add it but try to avoid flooding the main page by bumping too many questions on the topic.

I will soon create a new thread in order to discuss the ambiguities of some related tags I mentioned below.

This is not really an answer but far too long for a comment:

1. I don't think there's any other reason for the absence of than the evolution of the tags themselves.

2. Yes, having a looks like a good suggestion to me and its absence bothered me for quite some time already. I agree that all those questions you list (and many more) would benefit from the presence of such a tag.

At the moment I can think of the following tags with non-empty intersection with :

A while ago I or someone else eliminated the rarely used tag (one or two questions) because there seemed no need for that one.

I don't have a solution ready but I think the following should be discussed before action is taken:

I'm posting this answer as community-wiki so that others can add tags that should be discussed in this thread. Please do so.

• one slightly less than ideal, but nonetheless workable way of resolving the ambiguity between functional-analytic versus PDE notions of (operator-theory), (spectral-theory) and (harmonic-analysis) is to also tag the questions (pde) versus (functional-analysis). In the case of (harmonic-analysis) however, it may be best to just edit the tag wiki so that the (pde) style questions be re-directed to just (fourier-analysis) instead. – Willie Wong Jan 6 '12 at 9:09
• Actually, thinking about it again, I believe there's no reason not to introduce (banach-algebras) right now. The other ambiguities are a different matter and should probably be discussed elsewhere. Should I post the second part of my answer to the tag-merging and synonmys thread/open a new thread? – t.b. Jan 6 '12 at 9:37
• The eliminating of normed-algebras, which the post refers to, was discussed in comments to this question. // As for t.b.'s suggestion in the above comment; I think opening a new thread would be better than posting it in the long thread about tag-merging. – Martin Sleziak Jan 6 '12 at 9:39
• @t.b. it maybe wise to open a new thread, since comment fields are really horrible for discussion, that being doubly so on a BIG question like the tag-thread. – Willie Wong Jan 6 '12 at 9:42
• I am in favor of keeping the (operator-algebras) tag, although I don't have time to elaborate at the moment. – Jonas Meyer Jan 6 '12 at 14:23
• @Jonas: I was not suggesting to eliminate it. I was suggesting to use the more descriptive Banach-, $C^\ast$ and von Neumann-algebras tags whenever applicable (they could certainly live together with (operator-algebras)). Do you mean you're in favor of having operator-algebras instead of those three? – t.b. Jan 6 '12 at 14:27
• @t.b.: Oh, good. No, I want all of them to coexist. Sometimes Banach algebras are not operator algebras, and sometimes operator algebras are not C*- or von Neumann algebras (even though the latter isn't particularly common); for this and other reasons I think having all 4 is a good idea. – Jonas Meyer Jan 6 '12 at 14:47
• @Jonas: We seem to agree then; I clarified my last bullet point. I was merely making the point that I'm not 100% sure if the definition of (operator-algebras) as given on Wikipedia is sufficiently widely accepted while the three other ones have very well-defined meanings. – t.b. Jan 6 '12 at 14:58
• @t.b.: Yes, thank you. The Wikipedia definition is pretty inclusive, but not comprehensive, as there have been studies of algebras of unbounded operators. On the other hand, from my limited experience operator algebras are usually algebras of operators on Hilbert space, or normed (or matrix normed) algebras that have (completely) isometric representations on Hilbert space. The subject of operator algebras may also include the study of linear subspaces of $B(H)$, or normed (or matrix normed) spaces having (completely) isometric representations on Hilbert space (operator spaces).... – Jonas Meyer Jan 6 '12 at 19:32
• Or operator systems, ordered vector spaces having representations as self-adjoint subspaces of $B(H)$. Then again, in some contexts operator algebras refers exclusively to topics directly related to (abstract or concrete) C*- and von Neumann algebras, and the latter usage is I think the one most likely to be seen here. – Jonas Meyer Jan 6 '12 at 19:34
• I would go further and say that while the Wikipedia definition is not wrong, it seems to ignore how people actually use the term in practice. Historically operator algebras have meant C* and von Neumann algebras, with the prefix "nonselfadjoint" used if one works outside the *-closed setting; on the other hand, it is common and accepted to here the term used to mean any subalgebra of B(H). Subalgebras of B(X) are sometimes called Banach algebras of operators, or algebras of operators on Banach spaces, etc – user16299 Jan 6 '12 at 22:24
• @Jonas, .@Yemon: please see the update. Thanks for your clarifications on operator algebras. If one of you could work these comments into a tag-wiki for operator-algebras I would appreciate it. – t.b. Jan 7 '12 at 9:23
• @t.b.: I gave it a try. – Jonas Meyer Jan 7 '12 at 19:31
• @Jonas: This looks good to me, thank you. The only mild objection I have is the repetition of associated topics in two consecutive sentences. – t.b. Jan 7 '12 at 23:00
• @Jonas: Thanks, this looks much better. I wrote tag-wiki-excerpts for (banach-algebras) and (c-star-algebras). – t.b. Jan 8 '12 at 1:41