Type III factors with unique Cartan decomposition

Abstract

Abstract. We prove that for any free ergodic nonsingular nonamenable action Γ (X, µ) of all Γ in a large class of groups including all hyperbolic groups, the associated group measure space factor L ∞ (X) ⋊ Γ has L ∞ (X) as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in [PV12]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products M1 * B M2 over a subalgebra B of type I.

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

“…The next corollary strengthens the indecomposability properties of crossed product von Neumann algebras B ⋊ Γ arising from arbitrary actions Γ B of bi-exact discrete groups on abelian von Neumann algebras (see [Oz04,HV12,Is12] for previous results).…”

“…In this respect, our approach is similar to the one we developed in our previous paper [HI15]. We refer the reader to [HR14,HU15,HV12,Is12,Is13] for other structural/rigidity results for type III factors involving the continuous core decomposition.…”

Bernoulli Crossed Products Without Almost Periodic Weights

“…One of the main achievements in this direction was made by Popa, whose deformation/rigidity theory provided powerful classification results for type II 1 factors such as group measure space constructions L ∞ (X) ⋊ Γ of free ergodic probability measure preserving actions Γ (X, µ). Popa's deformation/rigidity theory has later been combined with modular theory to study type III factors, such as group measure space constructions for nonsingular actions [HV12], Shlyakhtenko's free Araki-Woods factors [Hou08] and free quantum group factors [Iso14]. In the same spirit, Stefaan Vaes and the author provided a classification result for noncommutative Bernoulli crossed products constructed with almost periodic states [VV14].…”

Unique prime factorization and bicentralizer problem for a class of type III factors

“…Popa introduced his powerful intertwining-by-bimodules techniques in[Po01] in the case when M is finite and more generally in[Po03] in the case when M is endowed with an almost periodic faithful normalstate ϕ ∈ M * for which 1 A , 1 B ∈ M ϕ , A ⊂ 1 A M ϕ 1 A and B ⊂ 1 B M ϕ 1 B .It was showed in [HV12, Ue12] that Popa's intertwining techniques extend to the case when B is semifinite and with expectation in 1 B M 1 B and A ⊂ 1 A M 1 A is any von Neumann subalgebra.In this section, we investigate a new setting in which A ⊂ 1 A M 1 A is any finite von Neumann subalgebra with expectation and B ⊂ 1 B M 1 B is any von Neumann subalgebra with expectation. This situation is technically more challenging than the one studied in[HV12,Ue12] since B can possibly be of type III and hence the basic construction M, B may no longer carry a faithful normal semifinite trace. Since we can no longer use the fact that B is semifinite as in [Ue12, Proposition 3.1], we use instead, as in the proof of [Po01, Theorem A.1], the canonical faithful normal semifinite operator valued weight from M, B to M and exploit the fact that A is a finite von Neumann algebra.…”