Timeline for Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Feb 8 at 13:22 | comment | added | Emil Jeřábek | @YCor Obliged. $ $ | |
| Feb 8 at 12:23 | answer | added | Mohammad Golshani | timeline score: 2 | |
| May 8, 2020 at 14:45 | comment | added | YCor | @LSpice "Stone-Cech corona" yields 20 times less Google occurences than "Stone-Cech remainder". One advantage of "Stone-Cech remainder" is that you can guess the meaning assuming that you know what "Stone-Cech compactification" is. I've actually encountered "corona" never in the meaning of this Wikipedia page (to which I'd recommend renaming), but in generalizations such such as the Higson-Roe corona, or binary corona of a metric space. | |
| May 8, 2020 at 14:36 | comment | added | LSpice | Although it's maybe less palatable these days, I think that the usual terminology for what you call the "Stone–Čech remainder" is the corona. | |
| Feb 25, 2020 at 19:04 | comment | added | Nik Weaver | Ah, I was mistaken as well --- as you seem to have guessed, I read "${\rm Aut}(S_\omega/{\rm fin})$" as "${\rm Aut}(P(\omega)/{\rm fin})$". | |
| Feb 25, 2020 at 17:17 | comment | added | YCor | @NikWeaver oops, sorry, actually $\mathrm{Aut}_{\mathrm{grp}}(S_\omega/\mathrm{fin})$ has cardinal $c$, not $2^c$ (which is not trivial by the way – Alperin-Covington-MacPherson/Truss). So my "immediate" argument fails in any case. Whether $\mathrm{Aut}_{\mathrm{BA}}(2^\omega/\mathrm{fin})$ embeds into $\mathrm{Aut}$(Calkin), I would have expected it too but without serious grounds. Sorry again! | |
| Feb 25, 2020 at 15:58 | comment | added | Nik Weaver | Is that why everyone has heard of Ilias's direction and no one has heard of mine? Because they assume my direction was trivial? Yikes. | |
| Feb 25, 2020 at 15:55 | comment | added | Nik Weaver | (Look again at whether ${\rm Aut}(S_\omega/{\rm fin})$ embeds in ${\rm Aut}(Q(l^2))$.) | |
| Feb 25, 2020 at 15:54 | comment | added | Nik Weaver | No, it's a lot harder for the Calkin algebra. Phillips and Weaver, The Calkin algebra has outer automorphisms, Duke Math. J. 139 (2007), 185–202. | |
| Feb 25, 2020 at 15:46 | comment | added | YCor | @NikWeaver But isn't existence of nontrivial outer automorphisms immediate from Rudin? Rudin proved under CH that $|\mathrm{Aut}(S_\omega/\mathrm{fin})|=2^c$. It immediately implies (since $\mathrm{Aut}(S_\omega/\mathrm{fin})$ embeds into $\mathrm{Aut}$(Calkin)) that $\mathrm{Aut}$(Calkin) has the same cardinality (and hence has Out of the same cardinality, since $\mathrm{Inn}$(Calkin) has cardinal $c$). | |
| Feb 25, 2020 at 15:43 | comment | added | Nik Weaver | Yes ... I'm aware of Ilias's paper. It followed a paper by Chris Phillips and me where we showed that CH implies outer automorphisms exist. Our techniques wouldn't be helpful for this problem though. | |
| Feb 25, 2020 at 15:40 | comment | added | YCor | @NikWeaver Nice question. Farah has results in this direction. For instance (Ann. Math. 2011 link) he proved the consistency of ZFC + all automorphisms are inner. In such a (quite exotic) model, they're non-conjugate since the Fredholm index distinguishes them. I don't know whether CH implies they're conjugate (if true this might be easier than the set-theoretic counterpart). | |
| Feb 25, 2020 at 14:29 | comment | added | Nik Weaver | YCor, have you looked at the analogous C*-algebra question: are the unilateral shift and its adjoint related by an automorphism of the Calkin algebra? Will Brian lists several facts about $\beta\omega\setminus\omega$ whose Calkin algebra analogues aren't familiar to me (but maybe experts would know better). | |
| Feb 18, 2020 at 16:20 | answer | added | Will Brian | timeline score: 16 | |
| Feb 18, 2020 at 15:49 | history | edited | Martin Sleziak |
added the (stone-cech-compactification) tag; if you don't consider it a good fit for the question, feel free to remove it
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| Feb 18, 2020 at 15:42 | history | asked | YCor | CC BY-SA 4.0 |