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

Abstract

Abstract. We consider crossed product II 1 factors M = N ⋊ σ G, with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ trace preserving actions of G on finite von Neumann algebras N that are "malleable" and mixing. Examples are the actions of G by Bernoulli shifts (classical and non-classical), and by Bogoliubov shifts. We prove a rigidity result for isomorphisms of such factors, showing the uniqueness, up to unitary conjugacy, of the position of the group von Neumann algebra L(G) inside M. We use this result to calculate the fundamental group of M, F (M), in terms of the weights of the shift σ, Introduction.This is the first of a series of papers in which we study rigidity properties of isomorphisms θ of crossed product II 1 factors M 0 , M arising from certain actions of groups on finite von Neumann algebras. We also study isomorphisms between amplifications of such factors. Typically, we assume the "source" factor M 0 comes from an action of a group G 0 having a large subgroup H ⊂ G 0 with the relative property (T) of Kazhdan-Margulis (G 0 is w-rigid), while the "target" factor M comes from an action (σ, G) with good "deformation+mixing" properties (a malleable action), e.g. an action by Bogoliubov or Bernoulli shifts (classical and non-classical). The "ideal" type of result we seek to prove, is that any isomorphism

Strongly solid group factors which are not interpolated free group factors

“…If τ (z) ≥ 1/n, since P z is diffuse amenable we may shrink z to z 0 ∈ P z such that τ (z 0 ) = 1/n and ctr P z (z 0 ) = cz 1 , where c is a scalar and z 1 ∈ Z(P z). Using Lemma 3.5 in [27], we get…”

“…Note that u * Qu = θ(Q). Moreover since θ(Q) is diffuse, Theorem 3.1 in [27] yields that the quasi-normalizer of θ(Q) inside…”

“…The proof of Theorem A, following a "deformation/rigidity" strategy, is a combination of the ideas and techniques in [16,21,27].…”

Kadison–Kastler stable factors

“…A key ingredient in this result is the vanishing of the bounded cohomology groups H 2 b .SL n .Z/; L 1 R .X; // for n 3 from [43], [4], and [45], and in Theorem A, which is then an immediate consequence of Theorem 6.10, SL n .Z/ can be replaced with any other group with this property. Via the work of [2] and [60]- [62], there are uncountably many pairwise nonisomorphic II 1 -factors to which this theorem applies (see Remark 6.12).…”

Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence

“…In this subsection, we briefly recall Popa's intertwiningby-bimodules [Po01,Po03]. In the present paper, we will need a generalization of Popa's intertwining-by-bimodules to the framework of type III von Neumann algebras developed by the authors in [HI15].…”