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We show that Shlyakhtenko's free Araki-Woods factors are strongly solid, meaning that for any diffuse amenable von Neumann subalgebra that is the range of a normal conditional expectation, the normalizer remains amenable. This provides the first class of nonamenable strongly solid type III factors. [ABSTRACT FROM AUTHOR]

In this article, we study a form of free transport for the interpolated free group factors, extending the work of Guionnet and Shlyakhtenko for the usual free group factors [11] . Our model for the interpolated free group factors comes from a canonical finite von Neumann algebra M ( Γ , μ ) associated to a finite, connected, weighted graph ( Γ , V , E , μ ) [12,13] . With this model, we use an operator-valued version of Voiculescu's free difference quotient introduced in [13] to state a Schwinger–Dyson equation which is valid for the generators of M ( Γ , μ ) . We construct free transport for appropriate perturbations of this equation. Also, M ( Γ , μ ) can be constructed using the machinery of Shlyakhtenko's operator-valued semicircular systems [24] . [ABSTRACT FROM AUTHOR]

We show that whenever m ≥ 1 and M 1 , … , M m are nonamenable factors in a large class of von Neumann algebras that we call C ( AO ) and which contains all free Araki–Woods factors, the tensor product factor M 1 ⊗ ‾ ⋯ ⊗ ‾ M m retains the integer m and each factor M i up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from [33,25] and moreover provides new UPF results in the case when M 1 , … , M m are free Araki–Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type I I I 1 factors in the class C ( AO ) . [ABSTRACT FROM AUTHOR]

We give a general description of the discrete decompositions of type III factors arising as central summands of free product von Neumann algebras based on our previous works. This enables us to give several precise structural results on type III free product factors. [ABSTRACT FROM AUTHOR]

We prove that the continuous version of the Connes-Shlyakhtenko first L2-cohomology for II1 factors, as proposed by Thom, always vanishes. [ABSTRACT FROM AUTHOR]

We show that the continuous core of any type III free product factor has no Cartan subalgebra. This is a complement to previous works due to Houdayer–Ricard and Boutonnet–Houdayer–Raum. [ABSTRACT FROM AUTHOR]

Consider a locally compact group G = Q ⋉ V such that V is abelian and the action of Q on the dual abelian group V ˆ has a free orbit of full measure. We show that such a group G can be quantized in three equivalent ways: (1) by reflecting across the Galois object defined by the canonical irreducible representation of G on L 2 (V) ; (2) by twisting the coproduct on the group von Neumann algebra of G by a dual 2-cocycle obtained from the G -equivariant Kohn–Nirenberg quantization of V × V ˆ ; (3) by considering the bicrossed product defined by a matched pair of subgroups of Q ⋉ V ˆ both isomorphic to Q. In the simplest case of the a x + b group over the reals, the dual cocycle in (2) is an analytic analogue of the Jordanian twist. It was first found by Stachura using different ideas. The equivalence of approaches (2) and (3) in this case implies that the quantum a x + b group of Baaj–Skandalis is isomorphic to the quantum group defined by Stachura. Along the way we prove a number of results for arbitrary locally compact groups G. Using recent results of De Commer we show that a class of G -Galois objects is parametrized by certain cohomology classes in H 2 (G ; T). This extends results of Wassermann and Davydov in the finite group case. A new phenomenon is that already the unit class in H 2 (G ; T) can correspond to a nontrivial Galois object. Specifically, we show that any nontrivial locally compact group G with group von Neumann algebra a factor of type I admits a canonical cohomology class of dual 2-cocycles such that the corresponding quantization of G is neither commutative nor cocommutative. [ABSTRACT FROM AUTHOR]