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When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group.

What is known about the following question: for which locally compact groups $G$ is $vN(G)$ a factor?

My understanding is that a complete characterisation isn't known - correct me if I'm wrong. What is it that makes the locally compact case so much harder than the discrete case?

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    $\begingroup$ 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. $\endgroup$
    – Onur Oktay
    Commented Jul 27, 2023 at 9:44
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    $\begingroup$ 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. $\endgroup$
    – Yemon Choi
    Commented Jul 27, 2023 at 11:05
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    $\begingroup$ 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. $\endgroup$
    – Yemon Choi
    Commented Jul 27, 2023 at 11:15
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    $\begingroup$ 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. $\endgroup$
    – Onur Oktay
    Commented Jul 27, 2023 at 12:01
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    $\begingroup$ @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. $\endgroup$
    – Yemon Choi
    Commented Jul 27, 2023 at 13:11

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