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

By Sorin Popa, S. Vaes. 2009

101 citations recorded by the source corpus.

We prove a “unique crossed product decomposition” result for group measure space II1 factors L ∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family $\mathcal{G}$, 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 $\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z})$ is W∗-superrigid, i.e. any isomorphism between L ∞(X)⋊Γ and an arbitrary group measure space factor L ∞(Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family $\mathcal{G}$, the Bernoulli actions of Γ are W∗-superrigid.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/cd166af80d56e36b36cf4afb80d2d7f077277303

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Citations in the corpus

W*-superrigidity for Bernoulli actions of property (T) groups (2010, methodology): “Note that Δ has been introduced in the proof of [42], Lemma 3.”
A CLASS OF SUPERRIGID GROUP VON NEUMANN ALGEBRAS (2010, background): “This gives its statement a W∗-superrigidity flavor, in the same spirit as the recent superrigidity results for group measure space II1 factors ([PV10], [Ioa11]), showing that certain classes of free ergodic probability measure preserving group actions G y (X,μ) can be completely recovered from their”
A class of groups for which every action is W$^*$-superrigid (2010, methodology): “X; / are shown in [PV09] to be W*-superrigid: any isomorphism between L1.”
Strong solidity of group factors from lattices in SO(n,1) and SU(n,1) (2010, background): “There is a well-known functor from the category whose objects are (separable) unitary representations of Γ and morphisms weak containment to the one of LΓ-LΓ correspondences and weak containment, cf. [22], which translates the representation theory of Γ into the theory of LΓ-LΓ correspondences. The ”
Orbit Equivalence and Measured Group Theory (2010, background): “A series of ground breaking results in von Neumann algebras obtained by Popa [Pop06a, Pop06c, Pop06d, Pop07a, Pop08] and his collaborators [PS07, IPP08, PV08d, PV08b, PV08a, Ioa08, PV08c, PV09b] (see [Vae07] for a review) dramatically modiﬁed the landscape.”
Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups (2012, methodology): “]), it would follow that β(2) n (Γ) are isomorphism invariants for arbitrary group measure space II1 factors L ∞(X) ⋊Γ. There is further supporting evidence for the above conjecture. For instance, in [PV09] we proved that a fairly large class of free product groups Γ = Γ1 ∗ Γ2, including all those w”