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A class of superrigid group von Neumann algebras. (English) Zbl 1295.46041

Rigidity phenomena in the von Neumann algebra world have a long story, tracing back to F. J. Murray and J. von Neumann [Ann. Math., Princeton, (2) 37, 116–229 (1936; JFM 62.0449.03)]. As a way to construct examples of von Neumann algebras, Murray and von Neumann used the group von Neumann algebra \(LG\) (that is, the \(W^*\)-algebra generated by the image of the left regular representation of \(G\)) and the group measure construction \(L^\infty(X)\rtimes G\). A very natural question is to what extent the von Neumann algebras remember the group \(G\). We refer to the introduction of the paper under review for many details of the story.
Over the last 15 years, rigidity results began to pile up. In the paper under review, the authors provide the first examples of \(W^*\)-superrigid icc groups, i.e., groups \(G\) such that their von Neumann algebra \(LG\) (a II\(_1\)-factor) has the property that \((LG)^t\simeq L\Lambda\) if and only if \(\Lambda\simeq G\) and \(t=1\). Groups with the property above are constructed as follows: start with a non-amenable group \(\Gamma_0\) and an amenable infinite group \(S\). Let \(\Gamma=\Gamma_0^{(S)}\rtimes S\) be the wreath product, \(I=\Gamma/S\), and let \(\Gamma\) act on \(I\) by left multiplication. For \(n\in\mathbb N\) square free, put \[ G=\left(\frac{\mathbb Z}{n\mathbb Z}\right)^{(\,I)}\rtimes\Gamma. \]
Plain wreath products are not \(W^*\)-superrigid, as the authors also show.
The techniques used follow the deformation/rigidity tradition in the area but, as the authors say, “we use the entire arsenal of ideas and techniques developed in previous papers” and the proofs require “more intricate deformation/rigidity arguments and a lot of technical effort”.

MSC:

46L10 General theory of von Neumann algebras
20F99 Special aspects of infinite or finite groups

Citations:

JFM 62.0449.03
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Full Text: DOI arXiv

References:

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