Bass-Serre rigidity results in von Neumann algebras

Abstract

We obtain new Bass-Serre type rigidity results for II1 equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any non-amenable factor arising as an amalgamated free product of von Neumann algebras M1 * B M2 over an abelian von Neumann algebra B, is prime, i.e. cannot be written as a tensor product of diffuse factors. This gives, both in the type II1 and in the type III case, new examples of prime factors.2000 Mathematics Subject Classification. 46L10; 46L54; 46L55.

Some OE- and W⁎-rigidity results for actions by wreath product groups

“…Further he showed that, given two stably orbit equivalent actions, σ and ρ, of such groups with σ | Γ and ρ| Λ ergodic, one has σ | Γ and ρ| Λ are stably orbit equivalent. He was also able to prove a similar measure equivalence rigidity for certain classes of direct products and amalgamated free products, thus obtaining rigidity results á la Monod and Shalom [20], as well as of Bass-Serre type [16,1,2]. His methods rely on Ozawa's techniques [23,24] involving the class S of groups, being C * -algebraic in nature and depending crucially on exactness of the groups involved.…”

“…Then we choose δ > 0 satisfying 1 − 2δ > √ 1 − c 2 and k sufficiently large such that for all u ∈ F we have u − θ t k (u) 2 6 .…”

“…By proceeding exactly as in the last part of the proof of Theorem 4.9 in [25], one derives that xζ 2 x 2 for all x ∈ M, [u, ζ ] 2 ε for all u ∈ F and ζ 2 δ. But then Corollary 2.3 in [25] shows that G is amenable.…”

Strongly solid group factors which are not interpolated free group factors

“…The following general result about intertwining subalgebras inside amalgamated free products will be a crucial tool in the next subsection (see also Theorem 4.2 in [4] and Theorem 5.6 in [14]). …”

Prime II1 factors arising from actions of product groups

“…Specifically, N. Ozawa showed that L ∞ (X) ⋊ Γ is prime whenever Γ (X, µ) is a free ergodic pmp action of a non-elementary hyperbolic group [Oz04] (see also [CS11]). By obtaining new Bass-Serre type rigidity results for II 1 factors, I. Chifan and C. Houdayer showed that the II 1 factor associated to any free ergodic pmp action of a free product group is prime [CH08]. Then by developing methods from [Si10,Va10b], D. Hoff proved that L ∞ (X) ⋊ Γ is prime whenever Γ (X, µ) is a free ergodic pmp action of a group which has positive first Betti number [Ho15].…”