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arXiv · 0906.2765

Group measure space decomposition of II_1 factors and W*-superrigidity

Sorin Popa·Stefaan Vaes
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Abstract

We prove a "unique crossed product decomposition" result for group measure space II_1 factors arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γin a fairly large family G, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T_n denotes the group of upper triangular matrices in PSL(n,Z), then any free, mixing p.m.p. action of the amalgamated free product of PSL(n,Z) with itself over T_n, is W*-superrigid, i.e. any isomorphism between L^\infty(X) \rtimes Γand an arbitrary group measure space factor L^\infty(Y) \rtimes Λ, comes from a conjugacy of the actions. We also prove that for many groups Γin the family G, the Bernoulli actions of Γare W*-superrigid.

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Sorin Popa, Stefaan Vaes. 2010-05-27. Group measure space decomposition of II_1 factors and W*-superrigidity. https://doi.org/10.1007/s00222-010-0268-5

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