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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.

##### MSC:
 46L35 Classifications of $$C^*$$-algebras
##### Keywords:
wreath products; relative property (T)
Full Text:
##### References:
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