Talk:Group algebra of a locally compact group

Section heading for L¹(G).

I suggest we remove the heading, and leave the content of that section so that it becomes part of the preceding section. CSTAR 14:51, 12 Oct 2004 (UTC)

Need simple definition for finite groups

ToDo: The very first part of this article should describe the group algebra for finite groups. linas 04:58, 9 Mar 2005 (UTC)

Never mind, I just added the beast. I needed something simple so that I could explain young symmetrizer. linas 06:39, 9 Mar 2005 (UTC)

Group C*-algebras

Now the group C*-algebras (maximal and reduced) are both defined twice.... --CSTAR 14:31, 29 October 2005 (UTC)[reply]

Yes ... I needed to think some more about that merge. Possibly there was little actually new in what I added. Charles Matthews 15:57, 29 October 2005 (UTC)[reply]

Group ring and group C*-algebra

Some of the material in this article (group algebra) should be relocated to group ring. A while back, the material on group C*-algebra was transferred from the group ring article to here. However, I think there should be a separate article on Group C*-algebra of a locally compact group. --CSTAR 00:30, 27 October 2006 (UTC)[reply]

Hopf and Frobenius structures

Any group algebra has the structure of a Hopf algebra and any group algebra for a finite group has the structure of a Frobenius algebra. Maybe someone should put that in. —Preceding unsigned comment added by 137.111.90.68 (talk) 02:07, 1 February 2008 (UTC)[reply]

Group algebra over topological groups

I tried to check the inclusion of the support of the convolution product. I get a condition on existence of elements of G that map

(Supp(g))^{-1}

to

Supp(f)

oh, it actually works, but one has to understand that we really have a product on the support!!! — Preceding unsigned comment added by 131.169.87.137 (talk • contribs) 15:00, 15 June 2013 (UTC)[reply]

Preamble

It doesn't seem good to me, if you don't mind, I would change it. I see two main shortcomings: group algebras are not necessarily Banach algebras, and the phrase about representations sounds too vague:

such that representations of the algebra are related to representations of the group

I would write something like

In functional analysis, group algebras are constructions that generalize the concept of group ring to some classes of topological groups with the aim to reduce the theory of representations of topological groups to the theory of representations of topological algebras. There are several main (non-equivalent) definitions of group algebra...

Eozhik (talk) 07:56, 31 August 2019 (UTC)[reply]

I did this. If nobody objects, I would make several other corrections to unify the presentation. Eozhik (talk) 13:30, 7 September 2019 (UTC)[reply]

(sorry for late response). You are correct that group algebras in generality are not necessarily Banach algebras. It’s just that the article limits itself to those cases. Should there be a more general treatment of group algebra? Perhaps, but we need a reference for that. —- Taku (talk) 00:35, 8 April 2020 (UTC)[reply]

TakuyaMurata, this sounds strange:

You are correct that group algebras in generality are not necessarily Banach algebras. It’s just that the article limits itself to those cases.

It is not written in the title of the article, even after your edits, that it is limited to Banach algebras. Where does this idea come from? And this:

Should there be a more general treatment of group algebra? Perhaps, but we need a reference for that.

The references for Banach group algebras don't disappear if we describe the algebraic case or give other definitions. What is the problem? Eozhik (talk) 04:44, 8 April 2020 (UTC)[reply]

The idea of Bananch algebra comes from the references the article use; it’s more common to use Banach algebras than more general topological algebras. In mathematics, almost always you can generalize the existing constructions. Mathematical possible does not mean editors are allowed to it; such a generalization needs to be well-established in literature before it can be discussed in Wikipedia. —— Taku (talk) 12:26, 9 April 2020 (UTC)[reply]

Removal of the section on group algebras of finite groups and other last edits

Dear TakuyaMurata, I don't agree with your words:

move idea section to stereotypical group algebra; as far as I can tell, this section is unrelated

What is meant here? Finite groups are special cases of locally compact groups. Their group algebras are special cases (and key examples) of group algebras. On the other hand, stereotype group algebra is one of the definitions of group algebra for locally compact groups. Why all this must be moved to another article? Eozhik (talk) 22:35, 7 April 2020 (UTC)[reply]

For example, that section emphasizes the aspect like a Hopf algebra structure or a category-theoretic universal property. Those aspects are important but, as I said, are somehow not relevant or not usually stressed in the discussion of group algebras in functional analysis (in the standard textbooks). —- Taku (talk) 00:33, 8 April 2020 (UTC)[reply]

TakuyaMurata group algebras are standard examples of Hopf algebras, and this is widely discussed in different sources, including textbooks. I do not see the necessity to limit the view of the reader to the cases when the proposed construction does not allow to prove this property. And this division into several articles, in which the property of being a Hopf algebra is only mentioned in the most exotic one, which in addition, has some prospect of being removed from Wikipedia, I find very strange. Eozhik (talk) 05:03, 8 April 2020 (UTC)[reply]

Of course, group algebras (of finite groups) are important examples of Hopf algebras. But this aspect is covered elsewhere in Wikipedia. But in functional analysis textbooks, you usually don’t see the discussion of Hopf algebras; they are more common in textbooks on algebraic groups or in algebraic topology. You said “I do not see the necessity to limit the view of the reader to the cases when the proposed construction does not allow to prove this property.” Mathematically, perhaps. But in Wikipedia the goal is not to pursue the mathematical elegant treatment of the topic but to pursue the best reflection of standard treatment of the topic. In other words, a Wikipedia article should look too different from a discussion in textbooks. This is why I moved “stereotype” type stuff to a separate article as that note does not appear in standard textbooks. This is why the article need to limit itself to Banach spaces (since that’s standard). —- Taku (talk) 11:35, 8 April 2020 (UTC)≐[reply]

TakuyaMurata, I suppose, here

a Wikipedia article should look too different from a discussion in textbooks

— you meant "a Wikipedia article should not look too different from a discussion in textbooks". Correct? Eozhik (talk) 11:46, 8 April 2020 (UTC)[reply]

Yes, sorry. "should" should have been "shouldn't". -- Taku (talk) 11:48, 8 April 2020 (UTC)[reply]

TakuyaMurata, so your point is that a Wikipedia article must describe only what is written in textbooks, right? Eozhik (talk) 12:04, 8 April 2020 (UTC)[reply]

Yes. Textbooks or some major monographs. nlab, on the other hand, is a place where it is ok to present mathematic ideals that are still not well established. -- Taku (talk) 12:11, 8 April 2020 (UTC)[reply]

TakuyaMurata, can you specify a rule in Wikipedia that you mean here? Eozhik (talk) 12:15, 8 April 2020 (UTC)[reply]

That would be Wikipedia:Reliable sources, Wikipedia:Notability and Wikipedia:OR. These policies are general policies but the common interpretation here to math articles is that you cannot present the theory or construction that is significantly beyond what is in textbooks, regardless of the mathematical soundness of such an extension . —- Taku (talk) 13:19, 8 April 2020 (UTC)[reply]

TakuyaMurata, I don't see this

you cannot present the theory or construction that is significantly beyond what is in textbooks, regardless of the mathematical soundness of such an extension

or anything equivalent to this, in these references. And this is a big news for me. If, as you say, this is a common attitude here, why isn't this stated directly in the rules? And how can I verify what you say, if this is not written? Eozhik (talk) 14:30, 8 April 2020 (UTC)[reply]

They are general policies and so you can’t find a rule like that explicitly; but the gist of those policy is that a Wikipedia is a place to present facts well established in literature. This is why papers authored by you cannot be enough; they may be reliable from the mathematical point of view but they are not reliable in the Wikipedia sense (as they are primary sources). Textbooks are not primary and so if the construction like stereotype group algebra can be found in textbooks, they can appear in Wikipedia; since that’s not the case, I removed it from here. I have also asked for opinions from other editors at Wikipedia_talk:WikiProject_Mathematics#Disambig_group_algebra so that we can be sure my interpretation of the policies is correct and common. -- Taku (talk) 14:49, 8 April 2020 (UTC)[reply]

TakuyaMurata, this is a very strong statement:

TakuyaMurata, so your point is that a Wikipedia article must describe only what is written in textbooks, right? Eozhik (talk) 12:04, 8 April 2020 (UTC)
Yes. Textbooks or some major monographs. ... -- Taku (talk) 12:11, 8 April 2020 (UTC)

It requires proof. Eozhik (talk) 15:00, 8 April 2020 (UTC)[reply]

The proof would be a common practice; in Wikipedia we routinely remove original research, i,e,, something not in textbooks. The removal is justified in the ground that Wikipedia is a place to present well-established facts.—— Taku (talk) 12:30, 9 April 2020 (UTC)[reply]

This is not original research:

The proof would be a common practice; in Wikipedia we routinely remove original research, i,e,, something not in textbooks.

This is a material of peer-reviewed scientific journals. And whether the facts are well-established, or not

The removal is justified in the ground that Wikipedia is a place to present well-established facts.

— is decided by editorial boards of these journals. If Wikipedia has its own understanding of this issue, this must be reflected in the local rules. Eozhik (talk) 15:00, 9 April 2020 (UTC)[reply]

It is indeed the common understanding in Wikipedia that a math article should be written mainly based on textbook references, instead of papers in a journal. It is of course fine and important to mention some ongoing research but the math articles themselves need not go too much beyond what can be found in widely-used textbooks. For math (and other scientific fields), textbooks differentiates what is considered most standard from what isn't. For example, it is possible to develop calculus based on the notion of infinitesimal but that's not the standard approach in standard textbooks on calculus, even if you can back up the infinitesimal theory based on (non-standard) calculus textbooks. The reason we need to limit ourselves to textbooks not what is mathematically possible is because, again, Wikipedia is a place to present well-established facts in the manner that is widely considered standard. Anyone, if competent, can do math research and can publish their results; that does not automatically mean they are part of mainstream mathematics. Usually textbooks are the best way to what is considered most standard. -- Taku (talk) 11:18, 10 April 2020 (UTC)[reply]

I don't see here evidence that this is a common understanding. I see only your understanding, TakuyaMurata. And this understanding contradicts to what I see here in Wikipedia, and to the usual encyclopedic traditions all over the world. There are lots of things reflected in encyclopedias without references to textbooks. And this is also solipsism. Eozhik (talk) 12:36, 10 April 2020 (UTC)[reply]

The evidence would be the style of other math articles; in Wikipedia, math articles are typically written based on materials from textbooks, *not* research papers. "what I see here in Wikipedia"; which math article is primary written based on research papers? To give just one example, group ring is clearly written based on textbooks. So are Banach algebra, Haar measures, etc. etc. Also, no encyclopedia including https://www.encyclopediaofmath.org/index.php/Main_Page has an entry on stereotype space. -- Taku (talk) 22:25, 11 April 2020 (UTC)[reply]

TakuyaMurata, your problem is that you don't check your statements. Here are some articles from the Soviet Mathematical Encyclopedia of 1977-1985 (later translated in Springer under the name “Encyclopedia of Mathematics”):

Fourier indices of an almost-periodic function

Heegaard decomposition

Homeomorphism group

Hypercomplex functions

Suzuki 2-group

Superharmonic function

Tertiary ideal

There are no references to textbooks in these articles. And among Wikipedia's articles on these topics only two, as I can see, namely Fubini's theorem and Tertiary ideal have such references. The articles Fano surface, Fréchet surface, Homeomorphism group, for example, don't. Good possibilities for you to destroy other people's work. Eozhik (talk) 13:05, 12 April 2020 (UTC)[reply]

There are also articles in “Encyclopedia of Mathematics” which initially (in the Russian edition) didn't have references to textbooks, but later, after translation these references were added (in "Comments"):

Urysohn-Brouwer lemma

Torelli theorems

Wallman compactification

This is normal for futher editions and doesn't cancel the work of the authors of Mathematical Encyclopedia. Eozhik (talk) 13:18, 12 April 2020 (UTC)[reply]

But they are not Wikipedia articles. To quote myself, "in Wikipedia, math articles are typically written based on materials from textbooks, *not* research papers." Also, the Encyclopedia of Mathematics often omit textbook references. Also, topics likes Fubini theorem are very well covered in textbooks. -- Taku (talk) 18:32, 12 April 2020 (UTC)[reply]

TakuyaMurata, you are not a mathematician. You do not understand the logical arguments, you do not follow the accuracy of the spoken and you do not hesitate to say absurd things:

But they are not Wikipedia articles.

Let us try to bet again? I claim that in my two posts, prior to your reply, there are references to Wikipedia articles. And this

Also, the Encyclopedia of Mathematics often omit textbook references.

— apparently must be understood as if initially those articles contained references to textbooks, but after translation those references were omited. That is not true, TakuyaMurata: there were no such references initially. And my point is that this is normal. There is no rule in human tradition to supply each encyclopedic article (on mathematics or not, this doesn't matter) with references to textbooks. You should read this attentively before replying next time. Eozhik (talk) 04:08, 13 April 2020 (UTC)[reply]

Wikipedia has its own rules; if you find those rules not normal then you need to find a consensus to change those rules. I said "But they are not Wikipedia articles." because I was referring to a common practice in Wikipedia; examples from the other encyclopedia do not make a strong refutation to my claim. —- Taku (talk) 19:17, 15 April 2020 (UTC)[reply]

Comment: In general, Taku is right that we shouldn’t stray too far from “standard” (in the sense of most reliable secondary/tertiary sources use) treatment. I don’t think that we should be over-reliant on textbooks and we should take care to adhere to WP:NOTTEXTBOOK, but at the same time the use of published secondary sources shouldn’t be based on anything that could be considered WP:FRINGE or not widely accepted.
I haven’t looked at the disputed content in this particular case, but it’s reasonable to impose some kind of mainstream view for an established topic like this.
@Eozhik: I have left you a note on your talk page about the personal attack above on TakuyaMurata. Please don’t make such ad hominem arguments nor personal statements about editors. — MarkH₂₁^talk 02:00, 15 April 2020 (UTC)[reply]

MarkH21, I replied. Eozhik (talk) 07:25, 15 April 2020 (UTC)[reply]

Section heading for L1(G).