Timeline for What is known about when $vN(G)$ is a factor, for a locally compact group $G$?
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13 events
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| Aug 7, 2023 at 11:19 | comment | added | Jared White | Thank you both. These references and insights are very helpful. | |
| Aug 4, 2023 at 2:55 | comment | added | Onur Oktay | Yet another reference about factor regular representations: please see Corollary 14.D.2 in the book by Bekka & de la Harpe arxiv.org/abs/1912.07262 | |
| Jul 27, 2023 at 18:20 | comment | added | Yemon Choi | @OnurOktay No problem! Like I said, I have made similar mistakes. This was just one case where I happened to know "countereexamples". | |
| Jul 27, 2023 at 14:27 | comment | added | Onur Oktay | @YemonChoi You're quite right, the paper assumes connectedness & simply connectedness. I tend to not share a reference if I didn't read & have an understanding of that reference. However, not so rarely I make crucial careless mistakes like this (getting old I guess). My apologies for the oversight, and thank you for straightening it up. | |
| Jul 27, 2023 at 13:13 | comment | added | Yemon Choi | BTW this shows the dangers of "picking results off the shelf" without checking what conventions are being followed - I have fallen into similar traps before! | |
| Jul 27, 2023 at 13:12 | history | edited | Jared White |
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| Jul 27, 2023 at 13:11 | comment | added | Yemon Choi | @OnurOktay Thank you for pointing out this paper. I am puzzled by the authors' claimed result, because as I said above I happen to know that there are solvable Lie groups for which VN(G)=B(H) which is definitely not type ${\rm II}_\infty$. Looking quickly at their paper, it seems that Corollary 4.12 only applies to connected and simply connected groups. This is not very clear in their paper, but see e.g. the remark before Proposition 4.1. | |
| Jul 27, 2023 at 12:01 | comment | added | Onur Oktay | Indeed there is a characterization for solvable Lie groups. Please see Theorem 4.9 in doi.org/10.48550/arXiv.2111.01034. By Corollary 4.12 in the same paper, those group VN-algebras are type $II_{\infty}$ factors. | |
| Jul 27, 2023 at 11:15 | comment | added | Yemon Choi | However, there are connected Lie groups whose vN algebra is isomorphic to B(H) and hence a Type I factor: for instance, take the "full real ax+b group", i.e. the semidirect product ${\bf R} \rtimes {\bf R}^\times$ where ${\bf R}^\times$ is the multiplicative group of ${\bf R}$ acting by dilations on the additive group of ${\bf R}$. I think that by replacing ${\bf R}$ with the p-adics ${\bf Q}_p$ one can get totally disconnected non-discrete examples. | |
| Jul 27, 2023 at 11:05 | comment | added | Yemon Choi | Hi Jared, I don't know of a good "big picture" answer to your final question, but it might be worth noting that if $G$ is a second-countable unimodular Type I group then the Plancherel formula/theorem can be interpreted as a disintegration of $VN(G)$ w.r.t. its centre, which then tends to be quite large for Lie examples. I would not be surprised if it turns out that $G$ second-countable+unimodular+VN(G) a factor implies $G$ totally disconnected. | |
| Jul 27, 2023 at 9:44 | comment | added | Onur Oktay | I do not know of a characterization, but you may find this article relevant & interesting: doi.org/10.1016/0022-1236(77)90073-8 constructs LC groups $G$ where $vN(G)$ are type III factors. | |
| S Jul 27, 2023 at 9:01 | review | First questions | |||
| Jul 27, 2023 at 9:23 | |||||
| S Jul 27, 2023 at 9:01 | history | asked | Jared White | CC BY-SA 4.0 |