Timeline for Abstract characterization of group von Neumann algebra (II1 factor)
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Nov 14, 2019 at 16:30 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
If A=C the action is not free.
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| Nov 14, 2019 at 16:08 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
Simplification of the question
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| Nov 14, 2019 at 10:17 | comment | added | YCor | But I'm really not a specialist in vN algebras. That a question has been solved recently doesn't make it off-topic. I upvoted the initial question because it was telling to me, and I think would be worth an answer by @Jiang (and, fine, along with its first variant solved by Boutonnet), maybe with just a reference, or with some further hints. | |
| Nov 14, 2019 at 9:41 | comment | added | Sebastien Palcoux | @YCor: You mean that you don't know whether there exist a $\mathrm{II}_1$ factor of the form $A⋊Γ$ (as above) but not of the form $LΓ$ or $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$ (as above)? Interesting! It could be an open problem also! I need to check that, perhaps it will be a future MathOverflow post... This subject admits so many open problems which can be "easily" formulated, but so hard to solve! | |
| Nov 14, 2019 at 9:20 | comment | added | YCor | The first one is simple. The second one is not exactly what you wrote (you wrote "to a group vN or a cross-product $L^\infty$... if not more generally stably iso..."). I'm not familiar with the topic enough to have a judgement on the question and see if it has an immediate counterexample as a variation of the previous ones. And even if we eventually reach an open question, the first two are real questions and are worth a full question-and-reply post. | |
| Nov 14, 2019 at 9:11 | comment | added | Sebastien Palcoux | @YCor The questions "Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism isomorphic to $LΓ$?" and "Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomorphic to $A⋊Γ$?" look almost as simple (to me) in their formulation. What is bothering you? "stably" or the cross-product? | |
| Nov 14, 2019 at 9:10 | comment | added | Sebastien Palcoux | @YCor I let Jiang choose, what should I do? | |
| Nov 14, 2019 at 0:25 | comment | added | YCor | I think that references to Ioana and Boutonnet would make a good answer, and that further questions could be asked separately (what I upvoted was the original question because of its simplicity, not its latest version). | |
| Nov 13, 2019 at 15:17 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
update again using Jiang comments
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| Nov 13, 2019 at 7:14 | comment | added | Jiang | It may be helpful to first find II$_1$ factors which are not stably isomorphic to any group factors or crossed products. | |
| Nov 13, 2019 at 7:04 | comment | added | Jiang | I do not know the answers. | |
| Nov 13, 2019 at 1:16 | comment | added | Sebastien Palcoux | @Jiang: Thanks! Again you wrote your answer as a comment instead of an answer. Should I update the question again? It is up to you. The next updated question should be: Is a $\mathrm{II}_1$ factor with an involutory antiautomorphism stably isomorphic to $LΓ$ or $L^{\infty}(X,\mu) \rtimes_{\sigma} \Gamma$, with $σ$ a pmp-action of an ICC group $Γ$ on a probability space $(X,μ)$, or more generally, stably isomorphic to $A⋊Γ$ with $A$ abelian von Neumann algebra? Do you also know the answer? | |
| Nov 12, 2019 at 21:31 | comment | added | Jiang | This was answered negatively again by theorem D in Remi Boutonnet's paper "W^*-superrigidity of mixing Gaussian actions of rigid groups". Indeed, the point is to find W^*-superrigid actions such that (Thm C in Ioana's paper holds) and the crossed product vn alg from this action is NOT a group vn alg. | |
| Nov 12, 2019 at 19:31 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
clarification that the question was updated after Jiang's comment
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| Nov 12, 2019 at 18:27 | comment | added | Sebastien Palcoux | @Jiang: Thanks! After Sakai, Connes, Jones and now Ioana, what should be the next updated question? Because you posted your answer as a comment and not as an answer, I understand that you implicitly agree that I update the question according to your comment. I just updated the question. Please let me know whether you appreciate and whether you also know the answer. | |
| Nov 12, 2019 at 18:25 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
updated question after Jiang comment
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| Nov 12, 2019 at 15:12 | comment | added | Jiang | The answer is negative as shown in Ioana's paper ``W*-superrigidity for Bernoulli actions of property (T) groups", see the discussion after corollary F there. | |
| Nov 12, 2019 at 12:26 | history | asked | Sebastien Palcoux | CC BY-SA 4.0 |