Talk:Amenable group

From PlanetMath
I simply took it from PlanetMath, let me know if it is not ok by some reason, they seem to hve GLP as well, but maybe a bit different flavor... Tosha 01:03, 16 May 2004 (UTC)

PlanetMath should be OK - as far as I know. Just add a link to the original article. Charles Matthews 08:34, 16 May 2004 (UTC)

Suggestions
It needs rewriting anyway..


 * Amenability makes sense for any toplogical group, and the notion is used in that generality


 * Missing is Folner's condition.

CSTAR 00:07, 17 May 2004 (UTC)


 * I agree, but I think it is already something to start from. Tosha 10:51, 19 May 2004 (UTC)

Soviet superiority
There are 12 or more conditions equivalent to amenability listed in an encyclopedia (the Soviet one) ... we should get writing ...

Charles Matthews 10:53, 19 May 2004 (UTC)

Simplified definition
I've added the simplified definition for discrete groups. There's a whole branch of math, geometric group theory, where groups are always discrete. We (geometric group theorists) totally ignore the possibility of topological groups; $$L^\infty$$ scares us.

''I'd like to see the article reorganized to emphasize that both definitions are equally legitimate in different contexts. Right now, the organization makes it look like the $$L^\infty$$ definition is the main one. (Although I agree it is more general than the discrete version.)'' Done.

Finally, my apologies if there's a mistake in the definition I added. I'll check it against a reference soon.

Dbenbenn 15:22, 4 Nov 2004 (UTC)


 * Apparently "soon" means "three months later". I found and corrected a small error.  Unfortunately, my reference is not currently citable.  I'll add the citation eventually.  dbenbenn | talk 00:43, 13 Feb 2005 (UTC)


 * It might be a good idea to refer to the "measure" in the second definition as a "finitely-additive measure" since most of the time, "measure" means countably-additive. It's confusing, since interesting amenable groups generally *do not* have countably-additive left-invariant probability measures.--Mattday 02:42, 26 September 2005 (UTC)

See also amenable algebra?
Should this be "amenable Banach algebra"? Also a red link, but a thing which is definitely defined AND very well connected to amenable groups. (A group is amenable [iff] the group Banach algebra $$L^1$$ is). A Geek Tragedy 15:53, 6 May 2007 (UTC) I made that article A Geek Tragedy 18:19, 25 May 2007 (UTC)

Pun?
I do not see the pun in the translation of M.M. Day. Admittedly, I am not a native English speaker, but so are many of the readers of Wikipedia. I think it should be explained. In either case, one should cite a source claiming that the translation is due to M.M. Day, and preferably expanding "M.M." part into full first and middle names. Boris Bukh (talk) 02:53, 17 April 2008 (UTC)
 * Fixed. Mahlon Marsh Day. Amenable group is one where you are *able* to find a *mean*, as in definition 1. See  for instance. JackSchmidt (talk) 03:07, 17 April 2008 (UTC)
 * Um. Was that the real reason, or an after-the-fact joke? The latter sounds a bit more plausible to me. By the way the pun doesn't work in American English (the middle syllable sounds like "men" rather than "mean"). --Trovatore (talk) 03:16, 17 April 2008 (UTC)

Firstly while editors may quibble over the use of the word "pun", let me remind them that the original paper of von Neumann was not cited in this article or the lead before I added it. Secondly, the myth of the "pun" is on mathematical record in many secondary sources. I have not been able to check the book of Greenleaf on Invariant Means, but for example

The term "amenable" was coined by M. Day about 1949 apparently as a pun ...
 * Bruce Blackadar writes in his article in Contemporary Mathematics Vol 365 (2004):

The term amenable was introduced by M.M. Day (as a pun).
 * In Paterson's book on amenability we read on Page 1

The first to use the adjective "amenable" was MM Day in [Day], apparently with a pun in mind. This is the abstract for an AMS meeting where the term was first introduced: [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183514222 Means on semigroups and groups, Bull. A.M.S. 55 (1949) 1054-1055]. Were any WP editors at this meeting? Mathsci (talk) 07:14, 17 April 2008 (UTC)
 * The same is said in Volker Runde's "Lectures on Amenability":


 * I will add link to this question on Mathematics StackExchange: Why is “Amenable Group” a pun? --Kompik (talk) 14:44, 15 March 2017 (UTC)

This is the worst pun in the history of the world. — Preceding unsigned comment added by 217.33.144.171 (talk) 15:22, 12 February 2018 (UTC)

Right-invariant
There seems to be a difference in the definitions of right-invariant and left-invariant (specifically, in the right-/left-actions).

$$L_g$$ is defined to be $$(L_gf)(h) = f(g^{-1}h)$$ while $$R_g$$ is defined to be $$(R_gf)(h) = f(hg)$$. Should it perhaps not be $$(R_gf)(h) = f(hg^{-1})$$? I'm not at work so I don't have any books in from of me to check but, well, that doesn't seem consistent...

Farpov (talk) 15:06, 30 May 2010 (UTC)

Tagging
expressed the view on my user talk page and at WP:SPI that the references, cited inline now, do not accord with what is in the article. I did not write this article. I have added to it significantly, keeping to the format of the original editors (I only have limited amounts of time for wikipedia). The views of Dricherby should have been expressed here, not on my user talk page. Perhaps he could please explain here which particular portions of the content are not in the references? Are there some problems of separability that concern him? (That is one point where I have been careful about what I have added.) He tagged the article after three edits by what appeared to be trolling by sockpuppets of the banned editor. The named account that edited this article was blocked indefinitely as a sockpuppet of Echigo mole, the troll sock community banned editor that apparently edited this article before him. Please could Dricherby try not to enable banned editors on wikipedia, particularly in specialist areas areas where he has no expertise at all? Thanks. Mathsci (talk) 13:12, 24 August 2012 (UTC)

Discrete group
A discrete group is equipped with the discrete topology, not with none at all. The references used to support this assertion in the article, namely, , , are not very useful without page references, so it is hard to say whether or not they support the assertion as made, but in any case it is incompatible with the Wikipedia article on the subject. Reak spoughly (talk) 14:39, 13 September 2012 (UTC)

Open problem since 1950
It is not clear what is means to say that the converse is "an open problem since 1950". Presumably it has always been open, and 1950 was the date on which it was first explicitly posed? Whatever it means, a reference would be helpful here. Reak spoughly (talk) 14:57, 13 September 2012 (UTC)
 * Sorry for the decade-long necromancy here, but I thought this was a very interesting point. In some Platonic sense, the question existed before 1950; it was just that no human mathematician had publicly posed it.  But in that same sense, it very likely had an answer, which simply was not known to human mathematicians, so perhaps it was not really "open".  In any case the "since 1950" text no longer appears in the article, so I suppose there's nothing for us to decide here. --Trovatore (talk) 19:44, 26 November 2022 (UTC)

Extension
An extension of amenable goups is amenable. Does this mean an extension of an amenable group by an amenable group? Is there a reference? Santander sorrow (talk) 08:27, 5 October 2012 (UTC)
 * I presume this is what is meant: it is certainly the case that an extension of an amenable group by an amenable group is again amenable. See for exmple Terry Tao's blog for 23 Jan 2010  or 14 Apr 2009 .  Do these count as Reliable sources?  Lichfielder (talk) 08:58, 5 October 2012 (UTC)

Axiom of choice nonsense
"The original definition, which depends on the axiom of choice." This makes no sense; a definition does not depend on an axiom.


 * Agreed. It should be removed or rephrased.  2001:171C:2E6F:B411:500F:5331:A8AF:90B9 (talk) 09:13, 2 July 2020 (UTC)

as a pun on... what?
To a non-native English speaker the pun is not obvious: a pun on what?

I'm supposing it is a pun on Mahlon Day's middlename initial ("M."): a"emm"nable. But that can't be right. Can boring translations of foreign (German) terms be technically classified as puns? To a NNES pun usually refers to a funny (joyful) wordplay, but messbar->amenable fails this criterion. Mdob (talk) 20:19, 17 November 2018 (UTC)
 * The idea is that you can put a mean on the group (a probability measure gives you the mean value of a function, by integrating), so it's a-mean-able. It doesn't really work in American English. --Trovatore (talk) 20:24, 17 November 2018 (UTC)


 * Thank you! I'll add that to the article. Mdob (talk) 16:33, 18 November 2018 (UTC)

Probable errors
"Note that A. Connes also proved that the von Neumann group algebra of any connected locally compact group is hyperfinite, so the last condition no longer applies in the case of connected groups."

Probably the stated condition is not equivalent, but a consequence of the other conditions. Otherwise, the Connes result would prove too much.

"Amenability is related to the spectral problem of Laplacians. For instance, the fundamental group of a closed Riemannian manifold is amenable if and only if the bottom of the spectrum of the Laplacian is 0 (R. Brooks, T. Sunada)."

Spectrum of the Laplacian on the universal cover?

2001:171C:2E6F:B411:500F:5331:A8AF:90B9 (talk) 09:13, 2 July 2020 (UTC)


 * I precised the statement of Brooks' theorem. (I don't know what Sunada proved on this topic.) Regarding your first point I don't understand what you are saying: that hyperfiniteness of the von Neumann algebra is not sufficient to guarantee amenability for discrete groups? jraimbau (talk) 12:12, 2 July 2020 (UTC)

"Amenability" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Amenability and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 November 26 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 10:46, 26 November 2022 (UTC)