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Cyril Houdayer

Publications and source records attributed to Cyril Houdayer.

At least 37 records · Page 2Linked to original sources
arXiv2018

Amenable absorption in amalgamated free product von Neumann algebras

We investigate the position of amenable subalgebras in arbitrary amalgamated free product von Neumann algebras $M = M_1 \ast_B M_2$. Our main result states that under natural analytic assumptions, any amenable subalgebra of $M$ that has a large intersection with $M_1$ is actually contained in $M_1$. The proof does not rely on Popa's asymptotic orthogonality property but on the study of non normal conditional expectations.

math.OA↗
arXiv2018

Structure of bicentralizer algebras and inclusions of type III factors

We investigate the structure of the relative bicentralizer algebra ${\rm B}(N \subset M, φ)$ for inclusions of von Neumann algebras with normal expectation where $N$ is a type ${\rm III_1}$ subfactor and $φ\in N_*$ is a faithful state. We first construct a canonical flow $β^φ: \mathbf R^*_+ \curvearrowright {\rm B}(N \subset M, φ)$ on the relative bicentralizer algebra and we show that the W$^*$-dynamical system $({\rm B}(N \subset M, φ), β^φ)$ is independent of the choice of $φ$ up to a canonical isomorphism. In the case when $N=M$, we deduce new results on the structure of the automorphism group of ${\rm B}(M,φ)$ and we relate the period of the flow $β^φ$ to the tensorial absorption of Powers factors. For general irreducible inclusions $N \subset M$, we relate the ergodicity of the flow $β^φ$ to the existence of irreducible hyperfinite subfactors in $M$ that sit with normal expectation in $N$. When the inclusion $N \subset M$ is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison's problem when $N$ is amenable.

math.OA↗
arXiv2017

Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras

We investigate factoriality, Connes' type ${\rm III}$ invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type ${\rm III}$ invariants.

math.OA↗
arXiv2017

Strongly ergodic equivalence relations: spectral gap and type III invariants

We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable $1$-cocycles with values into locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and $τ$ invariants for type ${\rm III}$ strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type ${\rm III_1}$ ergodic equivalence relation $\mathcal R$, the Maharam extension $\mathord{\text {c}}(\mathcal R)$ is strongly ergodic if and only if $\mathcal R$ is strongly ergodic and the invariant $τ(\mathcal R)$ is the usual topology on $\mathbf R$. We also obtain a structure theorem for almost periodic strongly ergodic equivalence relations analogous to Connes' structure theorem for almost periodic full factors. Finally, we prove that for arbitrary strongly ergodic free actions of bi-exact groups (e.g. hyperbolic groups), the Sd and $τ$ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

math.DS↗
arXiv2017

Singular MASAs in type III factors and Connes' Bicentralizer Property

We show that any type ${\rm III_1}$ factor with separable predual satisfying Connes' Bicentralizer Property (CBP) has a singular maximal abelian $\ast$-subalgebra that is the range of a normal conditional expectation. We also investigate stability properties of CBP under finite index extensions/restrictions of type ${\rm III_1}$ factors.

math.OA↗
arXiv2017

Strong solidity of free Araki-Woods factors

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.

math.OA↗
arXiv2017

Classification of a family of non almost periodic free Araki-Woods factors

We obtain a complete classification of a large class of non almost periodic free Araki-Woods factors $Γ(μ,m)"$ up to isomorphism. We do this by showing that free Araki-Woods factors $Γ(μ, m)"$ arising from finite symmetric Borel measures $μ$ on $\mathbf{R}$ whose atomic part $μ_a$ is nonzero and not concentrated on $\{0\}$ have the joint measure class $\mathcal C(\bigvee_{k \geq 1} μ^{\ast k})$ as an invariant. Our key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states. We use this to also deduce rigidity and classification theorems for free product von Neumann algebras.

math.OA↗
arXiv2017

Fullness and Connes' $\boldsymbolτ$ invariant of type III tensor product factors

We show that the tensor product $M \mathbin{\overline{\otimes}} N$ of any two full factors $M$ and $N$ (possibly of type ${\rm III}$) is full and we compute Connes' invariant $τ(M \mathbin{\overline{\otimes}} N)$ in terms of $τ(M)$ and $τ(N)$. The key novelty is an enhanced spectral gap property for full factors of type ${\rm III}$. Moreover, for full factors of type ${\rm III}$ with almost periodic states, we prove an optimal spectral gap property. As an application of our main result, we also show that for any full factor $M$ and any non-type ${\rm I}$ amenable factor $P$, the tensor product factor $M \mathbin{\overline{\otimes}} P$ has a unique McDuff decomposition, up to stable unitary conjugacy.

math.OA↗
arXiv2017

Rigidity of free product von Neumann algebras

Let $I$ be any nonempty set and $(M_i, φ_i)_{i \in I}$ any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class $\mathcal C_{\rm anti-free}$ of (possibly type III) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing a Cartan subalgebra. For the free product $(M, φ) = \ast_{i \in I} (M_i, φ_i)$, we show that the free product von Neumann algebra $M$ retains the cardinality $|I|$ and each nonamenable factor $M_i$ up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type II$_1$ factors and is new for free product type III factors. It moreover provides new rigidity phenomena for type III factors.

math.OA↗
arXiv2016

Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras

We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown. We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger-Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.

math.GR↗
arXiv2016

Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence

We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras $M = B \rtimes Γ$ arising from arbitrary actions $Γ\curvearrowright B$ 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 $N' \cap M^ω$ of any nonamenable von Neumann subalgebra with normal expectation $N \subset M$. We use this result to show that for any strongly ergodic essentially free nonsingular action $Γ\curvearrowright (X, μ)$ of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor ${\rm L}^\infty(X) \rtimes Γ$ has no nontrivial central sequence. Using recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15], we construct, for every $0 < λ\leq 1$, a type III$_λ$ strongly ergodic essentially free nonsingular action $\mathbf F_\infty \curvearrowright (X_λ, μ_λ)$ of the free group $\mathbf F_\infty$ on a standard probability space so that the corresponding group measure space type III$_λ$ factor ${\rm L}^\infty(X_λ, μ_λ) \rtimes \mathbf F_\infty$ 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.

math.OA↗
arXiv2016

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

We show that whenever $m \geq 1$ and $M_1, \dots, M_m$ are nonamenable factors in a large class of von Neumann algebras that we call $\mathcal C_{(\text{AO})}$ and which contains all free Araki-Woods factors, the tensor product factor $M_1 \mathbin{\overline{\otimes}} \cdots \mathbin{\overline{\otimes}} M_m$ 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 [OP03, Is14] and moreover provides new UPF results in the case when $M_1, \dots, M_m$ 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 ${\rm III_1}$ factors in the class $\mathcal C_{(\text{AO})}$.

math.OA↗
arXiv2016

Asymptotic structure of free product von Neumann algebras

Let $(M, φ) = (M_1, φ_1) \ast (M_2, φ_2)$ be the free product of any $σ$-finite von Neumann algebras endowed with any faithful normal states. We show that whenever $Q \subset M$ is a von Neumann subalgebra with separable predual such that both $Q$ and $Q \cap M_1$ are the ranges of faithful normal conditional expectations and such that both the intersection $Q \cap M_1$ and the central sequence algebra $Q' \cap M^ω$ are diffuse (e.g. $Q$ is amenable), then $Q$ must sit inside $M_1$. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion $M_1 \subset M$ in arbitrary free product von Neumann algebras.

math.OA↗
arXiv2015

Free independence in ultraproduct von Neumann algebras and applications

The main result of this paper is a generalization of Popa's free independence result for subalgebras of ultraproduct ${\rm II_1}$ factors [Po95] to the framework of ultraproduct von Neumann algebras $(M^ω, φ^ω)$ where $(M, φ)$ is a $σ$-finite von Neumann algebra endowed with a faithful normal state satisfying $(M^φ)' \cap M = \mathbf{C} 1$. More precisely, we show that whenever $P_1, P_2 \subset M^ω$ are von Neumann subalgebras with separable predual that are globally invariant under the modular automorphism group $(σ_t^{φ^ω})$, there exists a unitary $v \in \mathcal U((M^ω)^{φ^ω})$ such that $P_1$ and $v P_2 v^*$ are $\ast$-free inside $M^ω$ with respect to the ultraproduct state $φ^ω$. Combining our main result with the recent work of Ando-Haagerup-Winsløw [AHW13], we obtain a new and direct proof, without relying on Connes-Tomita-Takesaki modular theory, that Kirchberg's quotient weak expectation property (QWEP) for von Neumann algebras is stable under free product. Finally, we obtain a new class of inclusions of von Neumann algebras with the relative Dixmier property.

math.OA↗
arXiv2015

Asymptotic structure of free Araki-Woods factors

The purpose of this paper is to investigate the structure of Shlyakhtenko's free Araki-Woods factors using the framework of ultraproduct von Neumann algebras. We first prove that all the free Araki-Woods factors $Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ are $ω$-solid in the following sense: for every von Neumann subalgebra $Q \subset Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ that is the range of a faithful normal conditional expectation and such that the relative commutant $Q' \cap M^ω$ is diffuse, we have that $Q$ is amenable. Next, we prove that the continuous cores of the free Araki-Woods factors $Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ associated with mixing orthogonal representations $U : \mathbb R \to \mathcal O(H_{\mathbb R})$ are $ω$-solid type ${\rm II_\infty}$ factors. Finally, when the orthogonal representation $U : \mathbb R \to \mathcal O(H_{\mathbb R})$ is weakly mixing, we prove a dichotomy result for all the von Neumann subalgebras $Q \subset Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ that are globally invariant under the modular automorphism group $(σ_t^{φ_U})$ of the free quasi-free state $φ_U$.

math.OA↗
arXiv2015

Gamma stability in free product von Neumann algebras

Let $(M, φ) = (M_1, φ_1) \ast (M_2, φ_2)$ be a free product of arbitrary von Neumann algebras endowed with faithful normal states. Assume that the centralizer $M_1^{φ_1}$ is diffuse. We first show that any intermediate subalgebra $M_1 \subset Q \subset M$ which has nontrivial central sequences in $M$ is necessarily equal to $M_1$. Then we obtain a general structural result for all the intermediate subalgebras $M_1 \subset Q \subset M$ with expectation. We deduce that any diffuse amenable von Neumann algebra can be concretely realized as a maximal amenable subalgebra with expectation inside a full nonamenable type III$_1$ factor. This provides the first class of concrete maximal amenable subalgebras in the framework of type III factors. We finally strengthen all these results in the case of tracial free product von Neumann algebras.

math.OA↗
arXiv2015

Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity

We introduce an algebraic invariant for aperiodic inclusions of probability measure preserving equivalence relations. We use this invariant to prove that every stable orbit equivalence between free pmp actions of direct products of non-amenable Baumslag-Solitar groups whose canonical subgroup acts aperiodically forces the number of factors of the products to be the same and the factors to be isomorphic after permutation. This generalises some of the results obtained by Kida and moreover provides new measure equivalence rigidity phenomena for Baumslag-Solitar groups. We also obtain a complete classification of direct products of relative profinite completions of Baumslag-Solitar groups, continuing recent work of Elder and Willis.

math.GR↗