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Rigidity results for wreath product \(\mathrm{II}_1\) factors. (English) Zbl 1134.46041
Summary: We consider \(\text{II}_1\) factors of the form \(M=(\overline{\bigotimes}_G\, B)\rtimes G\), where either (i) \(B\) is a non-hyperfinite \(\text{II}_1\) factor and \(G\) is an ICC amenable group, or (ii) \(B\) is a weakly rigid \(\text{II}_1\) factor and \(G\) is an ICC group acting on \(\overline{\bigotimes}_G\, B\) by Bernoulli shifts. We prove that isomorphy of two such factors implies cocycle conjugacy of the corresponding Bernoulli shift actions. In particular, the groups acting must be isomorphic. As a consequence, we can distinguish between certain classes of group von Neumann algebras associated to wreath product groups.

46L35 Classifications of \(C^*\)-algebras
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