Group measure space decomposition of II₁ factors and W*-superrigidity

الأصل · EN

We prove a "unique crossed product decomposition" result for group measure space II₁ 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 Tₙ 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 Tₙ, is W*-superrigid, i.e. any isomorphism between L∞(X) Γand an arbitrary group measure space factor L∞(Y) 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.