Cyril Houdayer

A class of ₁ factors with an exotic abelian maximal amenable subalgebra

We show that for every mixing orthogonal representation π : Z → O(HR), the abelian subalgebra L(Z) is maximal amenable in the crossed product II1 factor Γ(HR) ′′ ⋊π Z associated with the free Bogoljubov action of the representation π. This provides uncountably many non-isomorphic A-A-bimodules which are disjoint from the coarse A-A-bimodule and of the form L(M ⊖ A) where A ⊂ M is a maximal amenable masa in a II1 factor.

A class of groups for which every action is W*-superrigid

We prove the uniqueness of the group measure space Cartan subalgebra in crossed products A rtimes Gamma covering certain cases where Gamma is an amalgamated free product over a non-amenable subgroup. In combination with Kidas work we deduce that if Sigma < SL(3,Z) denotes the subgroup of matrices g with g_31 = g_32 = 0, then any free ergodic probability measure preserving action of Gamma = SL(3,Z) *_Sigma SL(3,Z) is stably W*-superrigid. In the second part we settle a technical issue about the unitary conjugacy of group measure space Cartan subalgebras.

Amalgamated free product type III factors with at most one Cartan subalgebra

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras M1 * B M2 over an amenable von Neumann subalgebra B. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra (M1, ϕ1) * (M2, ϕ2) with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in [Io12a]. Next, we prove that any countable nonsingular ergodic equivalence relation R defined on a standard measure space and which splits as the free product R = R1 * R2 of recurrent subequivalence relations gives rise to a nonamenable factor L(R) with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors L ∞ (X) ⋊ Γ arising from nonsingular free ergodic actions Γ (X, µ) on standard measure spaces of amalgamated groups Γ = Γ1 * Σ Γ2 over a finite subgroup Σ.

Structural results for free Araki–Woods factors and their continuous cores

We show that for any type

Gamma Stability in Free Product von Neumann Algebras

{rm III_1}

Strong solidity of free Araki-Woods factors

free Araki-Woods factor

AMENABLE ABSORPTION IN AMALGAMATED FREE PRODUCT VON NEUMANN ALGEBRAS

mathcal{M} = Gamma(H_R, U_t)

Free independence in ultraproduct von Neumann algebras and applications

associated with an orthogonal representation

Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors

(U_t)

TYPE III FACTORS WITH UNIQUE CARTAN DECOMPOSITION

of