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

Abstract

Abstract. We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras M = B ⋊Γ arising from arbitrary actions Γ B of bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. We prove a spectral gap rigidity result for the central sequence algebra N ′ ∩ M ω of any nonamenable von Neumann subalgebra with normal expectation N ⊂ M . We use this result to show that for any strongly ergodic essentially free nonsingular action Γ (X, µ) of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor L ∞ (X) ⋊ Γ has no nontrivial central sequence. Using recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15], we construct, for every 0 < λ ≤ 1, a type III λ strongly ergodic essentially free nonsingular action F∞ (X λ , µ λ ) of the free group F∞ on a standard probability space so that the corresponding group measure space type III λ factor L ∞ (X λ , µ λ ) ⋊ F∞ has no nontrivial central sequence by our main result. In particular, we obtain the first examples of group measure space type III factors with no nontrivial central sequence.

Strongly ergodic equivalence relations: spectral gap and type III invariants

“…Any nonamenable countable discrete group Γ admits a strongly ergodic free pmp action, namely the plain Bernoulli action Γ ([0, 1] Γ , Leb ⊗Γ ). We refer the reader to [HI15,Oz16] and the references therein for various examples of type III strongly ergodic free actions associated with free groups ( [HI15]) and lattices in simple connected Lie groups with finite center ( [Oz16]).…”

A remark on fullness of some group measure space von Neumann algebras

“…Indeed, by [HI16,Lemma 5.1], if M 0 were not full, then there would be a unitary central sequence (w n ) n in M 0 such that t∈F |w t n | 2 → 0 ultrastrongly for every finite subset F ⊂ Λ. But then Theorem 3, applied to Ω = Λ\Λ nac (note that we may assume that (X 0 , µ 0 ) is non-atomic and thus the Λ-action on it is non-amenable), implies that 1 M 0 = t∈Λ |w t n | 2 → 0, which is absurd.…”

“…Recall that a von Neumann factor N is said to be full [Con74] if the central sequence algebra N ∩ N ω is trivial for a non-principal ultrafilter ω. For more information on fullness, we refer the reader to [AH14,HI16] and the references therein. Recently, Houdayer and Isono [HI16] have studied which group has the property that the group measure space factor Γ L ∞ (X) is full for every non-singular strongly ergodic essentially free action Γ (X, µ) on a standard measure space, and they have proved, among other things, that biexact groups (e.g., hyperbolic groups) have this property.…”

“…For more information on fullness, we refer the reader to [AH14,HI16] and the references therein. Recently, Houdayer and Isono [HI16] have studied which group has the property that the group measure space factor Γ L ∞ (X) is full for every non-singular strongly ergodic essentially free action Γ (X, µ) on a standard measure space, and they have proved, among other things, that biexact groups (e.g., hyperbolic groups) have this property. Recall that a non-singular action Γ (X, µ) on a probability space (if µ is not a probability measure, replace it with a probability measure in the same measure class) is said to be strongly ergodic if any sequence (E n ) n of measurable subsets such that µ(E n sE n ) → 0 for every s ∈ Γ has to be trivial: µ(E n )(1 − µ(E n )) → 0.…”

Structure of modular invariant subalgebras in free Araki–Woods factors

“…Next, fix ω ∈ β(N) \ N any nonprincipal ultrafilter. By [HR14, Theorem A] (see also [HI15,Theorem 7.1]), we have that (Q ′ ∩ M ω )z ⊥ = (Q ′ ∩ M )z ⊥ is atomic.…”