Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II

By Sorin Popa. 2003

125 citations recorded by the source corpus.

We prove that any isomorphism θ:M0≃M of group measure space II1 factors, $M_0=L^\infty(X_0, \mu_0) \rtimes_{\sigma_0} G_0$, $M=L^\infty(X, \mu)\rtimes_{\sigma} G$, with G0 an ICC group containing an infinite normal subgroup with the relative property (T) of Kazhdan-Margulis (i.e. G0w-rigid) and σ a Bernoulli action of some ICC group G, essentially comes from an isomorphism of probability spaces which conjugates the actions with respect to some identification G0≃G. Moreover, any isomorphism θ of M0 onto a “corner” pMp of M, for p∈M an idempotent, forces p=1. In particular, all group measure space factors associated with Bernoulli actions of w-rigid ICC groups have trivial fundamental group and any isomorphism of such factors comes from an isomorphism of the corresponding groups. This settles a “group measure space version” of Connes rigidity conjecture, shown in fact to hold true in a greater generality than just for ICC property (T) groups. We apply these results to ergodic theory, establishing new strong rigidity and superrigidity results for orbit equivalence relations.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a4dfce3d11786e958beb246e8c1f91f32095832

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Citations in the corpus

On Ozawa's Property for Free Group Factors (2006, unknown): “By ([P1], or 2.1 in [ P3 ], or [PSiSm]; a modification of arguments in [Ch] will also do), there exist projections q ∈ Q 0 , q ′ ∈ (Q 0 ) ′ ∩ M, p ∈ αt(N), an isomorphism ρ : qQ 0 q → pαt(N)p, a projection p ′ ∈ ρ(qQ 0 qq ′ ) ′ ∩ pMp and a partial isometry v ∈ M such that v ∗ v = qq ′ , vv ∗ = pp ′ ”
RELATIVE KAZHDAN PROPERTY (2005, methodology): “or the pair (SL2(R)⋉R2,R2). The deﬁnition of relative Property (T) has been extended in [HV] to pairs (G,H) with Hnot necessarily normal in G. Such pairs are extensively used in the work of Popa (see [Popa] and the references therein), in the context of operator algebras. This motivated, for instanc”
Rigidity results for wreath product II1 factors (2006, methodology): “Recall from [ Po2 ] that an action � of a discrete group G on a finite von Neumann algebra N is called (tracial) malleable if there exists finite von Neumann algebra ˜”
W*‐superrigidity for group von Neumann algebras of left–right wreath products (2012, methodology): “Contrary to the approach of [13], our proof does not use the clustering techniques of [23], but uses the recent results of [10, 28] instead.”
Orbit Equivalence and Measured Group Theory (2010, unknown): “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.”
Group measure space decomposition of II1 factors and W*-superrigidity (2009, methodology): “By using intertwining subalgebras techniques [44, 45 ], in order to prove the uniqueness, up to unitary conjugacy, of the Cartan subalgebra A = L ∞ (X) of a group”
Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups (2006, methodology): “The first von Neumann strong rigidity theorem, deducing conjugacy of actions out of their von Neumann equivalence, was recently obtained by the first author [ 34 , 35], who proved that von Neumann equivalence of an essentially free action of a w-rigid group on the one hand and the Bernoulli action o”
Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups (2005, methodology): “The proof is very similar to (4.2 in [P1]), which served as a model for ([ P2 ,3], [PSa]) as well.”