Timeline for Why are abelian groups amenable?
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 3, 2023 at 16:48 | comment | added | Vivaan Daga | Can't you use the Hahn-Banach theorem to get a left-invariant mean? | |
| Jan 9, 2013 at 9:56 | answer | added | Freek_H | timeline score: 4 | |
| Apr 19, 2011 at 18:05 | answer | added | Alain Valette | timeline score: 12 | |
| Jan 20, 2010 at 0:31 | vote | accept | Tom Leinster | ||
| Jan 19, 2010 at 5:11 | answer | added | Tom Church | timeline score: 31 | |
| Jan 18, 2010 at 22:27 | history | edited | Tom Leinster | CC BY-SA 2.5 |
Big update
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| Jan 18, 2010 at 21:53 | comment | added | Tom Leinster | Right, Mariano, I feel the same way. | |
| Jan 18, 2010 at 19:36 | answer | added | Tom Church | timeline score: 4 | |
| Jan 18, 2010 at 4:14 | comment | added | Yemon Choi | I agree with Mariano, I think. Is it a bit like saying that the product in Set of a family of non-empty objects is non-empty? There are, especially in functional analysis, so many plausible-sounding and useful statements which turn out to (more or less) need AC, and some of them even sound categorical enough that one might not get flamed for using them ;) | |
| Jan 18, 2010 at 4:09 | comment | added | Mariano Suárez-Álvarez | "More natural" means mostly "I have trained myself to think it is natural that an arbitrarily humogous product of intervals is compact", I guess. | |
| Jan 18, 2010 at 4:09 | comment | added | JBorger | I know nothing about these things, but here are some general remarks (which are probably not news to most people). It seems to me that if you need the axiom of choice for even the simplest of groups, like Z, then without refining your definitions, you shouldn't be surprised if every little result becomes a long, hard slog. An obvious attempt to repair this would be to find a version of amenability which (i) is equivalent assuming the axiom of choice and (ii) holds for Z without assuming the axiom of choice. | |
| Jan 18, 2010 at 4:05 | answer | added | Yemon Choi | timeline score: 9 | |
| Jan 18, 2010 at 3:51 | comment | added | Tom Leinster | Thanks, Mariano. I'll have to take your word for most of that. Why do you say "more natural"? | |
| Jan 18, 2010 at 3:36 | comment | added | Mariano Suárez-Álvarez | Can't one establish amenability of $\mathbb Z$ by using the second characterization of amenability given in wikipedia (Γ is amenable iff whenever Γ acts by isometries on a (separable) Banach space E, leaving a weakly closed convex subset C of the closed unit ball of E* invariant, then Γ has a fixed point in C) and the Ryll-Nardzewski fixed point theorem? This also needs Choice in order to prove Tychonof in order to prove the Banach–Alaoglu theorem, yet it seems more natural... | |
| Jan 18, 2010 at 3:15 | comment | added | Tom Leinster | Thanks mathphysicist. Fixed. (I wrote "every abelian group is abelian", which even I can prove.) | |
| Jan 18, 2010 at 3:05 | comment | added | mathphysicist | In #6, there appears to be a typo. Shouldn't the beginning of #6 read "every abelian group is amenable"? | |
| Jan 18, 2010 at 3:02 | history | asked | Tom Leinster | CC BY-SA 2.5 |