arXiv subjects

Cyril Houdayer

Publications and source records attributed to Cyril Houdayer.

At least 19 recordsLinked to original sources

arXiv2026

The classification of flows on $\mathrm{II}_1$ factors and Connes' bicentralizer problem

We settle two long-standing open problems in von Neumann algebras. First, we show that every outer flow with full Connes spectrum on the hyperfinite $\mathrm{II}_1$ factor has the Rokhlin property. By the work of Masuda and Tomatsu, such a flow is therefore unique up to cocycle conjugacy. This settles Takesaki's classification problem for flows on the hyperfinite type $\mathrm{II}_1$ factor. Drawing on type $\mathrm{III}$ theory, we develop a bicentralizer machinery for trace-preserving actions of locally compact groups. In the amenable case, we relate the bicentralizer conjecture to the Rokhlin property. For abelian groups, we prove an analog of Connes-Størmer transitivity theorem and we generalize Connes-Takesaki relative commutant theorem. A new resonance phenomenon is revealed which allows us to solve the bicentralizer conjecture for actions of $\R$. We then go back to the type $\mathrm{III}$ world and use this new resonance phenomenon to solve Connes' bicentralizer conjecture for all type $\mathrm{III}_1$ factors.

math.OA↗

arXiv2026

The noncommutative topological factor theorem for rank-one product lattices

We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that the corresponding classification for the full flag action of $\operatorname{SL}_3(\mathbb Z)$ would imply ordinary ITAP.

math.OA↗

arXiv2026

Weyl groups and rigidity of von Neumann algebras

Let $G$ be a noncompact semisimple algebraic group with trivial center, $S < G$ a maximal split torus, $H < G$ the centralizer of $S$ in $G$ and $Γ< G$ an irreducible lattice. Consider the group measure space von Neumann algebra $\mathscr M = \operatorname{L}(Γ\curvearrowright G/H)$ associated with the nonsingular action $Γ\curvearrowright G/H$ and regard the group von Neumann algebra $M = \operatorname{L}(Γ)$ as a von Neumann subalgebra $M \subset \mathscr M$. We show that the group $\operatorname{Aut}_M(\mathscr M)$ of all unital normal $\ast$-automorphisms of $\mathscr M$ acting identically on $M$ is isomorphic to the Weyl group $\mathscr W_G$ of the semisimple algebraic group $G$. Our main theorem is a noncommutative analogue of a rigidity result of Bader-Furman-Gorodnik-Weiss for group actions on algebraic homogeneous spaces and moreover gives new insight towards Connes' rigidity conjecture for higher rank lattices.

math.OA↗

arXiv2026

Uniqueness of almost periodic outer flows on the hyperfinite type $\mathrm{II}_1$ factor

We show that any almost periodic outer flow $α: \mathbb R \curvearrowright R$ on the hyperfinite type $\mathrm{II}_1$ factor with Connes' spectrum $Γ(α) = \mathbb R$ satisfies the Rokhlin property and thus is unique up to cocycle conjugacy. The proof relies on a key cocycle perturbation result for type $\mathrm{III}$ amenable equivalence relations. As a byproduct of our methods, we also show that every almost periodic factor of type $\mathrm{III}_1$ with separable predual has an extremal almost periodic faithful normal state.

math.OA↗

arXiv2026

Selfless W$^*$-probability spaces and Connes' bicentralizer problem

We introduce the notion of selfless W$^*$-probability space and study its connection with Connes' bicentralizer problem. In particular, we show that if $M$ is a separable type ${\rm III_1}$ factor with trivial bicentralizer, then $(M, φ)$ is selfless for every faithful normal state $φ\in M_\ast$.

math.OA↗

arXiv2025

Strong primeness for equivalence relations arising from Zariski dense subgroups

We show that orbit equivalence relations arising from essentially free ergodic probability measure preserving actions of Zariski dense discrete subgroups of simple algebraic groups are strongly prime. As a consequence, we prove the existence and the uniqueness of a prime factorization for orbit equivalence relations arising from direct products of higher rank lattices. This extends and strengthens Zimmer's primeness result for equivalence relations arising from actions of lattices in simple Lie groups. The proof of our main result relies on a combination of ergodic theory of algebraic group actions and Popa's intertwining theory for equivalence relations.

math.DS↗

arXiv2025

Operator algebraic characterization of the noncommutative Poisson boundary

We obtain an operator algebraic characterization of the noncommutative Furstenberg-Poisson boundary $\operatorname{L}(Γ) \subset \operatorname{L}(Γ\curvearrowright B)$ associated with an admissible probability measure $μ\in \operatorname{Prob}(Γ)$ for which the $(Γ, μ)$-Furstenberg-Poisson boundary $(B, ν_B)$ is uniquely $μ$-stationary. This is a noncommutative generalization of Nevo-Sageev's structure theorem [NS11]. We apply this result in combination with previous works to provide further evidence towards Connes' rigidity conjecture for higher rank lattices.

math.OA↗

arXiv2023

Asymptotic freeness in tracial ultraproducts

We prove novel asymptotic freeness results in tracial ultraproduct von Neumann algebras. In particular, we show that whenever $M = M_1 \ast M_2$ is a tracial free product von Neumann algebra and $u_1 \in \mathscr U(M_1)$, $u_2 \in \mathscr U(M_2)$ are Haar unitaries, the relative commutants $\{u_1\}' \cap M^{\mathcal U}$ and $\{u_2\}' \cap M^{\mathcal U}$ are freely independent in the ultraproduct $M^{\mathcal U}$. Our proof relies on Mei-Ricard's results [MR16] regarding $\operatorname{L}^p$-boundedness (for all $1 < p < +\infty$) of certain Fourier multipliers in tracial amalgamated free products von Neumann algebras. We derive two applications. Firstly, we obtain a general absorption result in tracial amalgamated free products that recovers several previous maximal amenability/Gamma absorption results. Secondly, we prove a new lifting theorem which we combine with our asymptotic freeness results and Chifan-Ioana-Kunnawalkam Elayavalli's recent construction [CIKE22] to provide the first example of a ${\rm II_1}$ factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras.

math.OA↗

arXiv2023

Charmenability of higher rank arithmetic groups

We complete the study of characters on higher rank semisimple lattices initiated in [BH19,BBHP20], the missing case being the case of lattices in higher rank simple algebraic groups in arbitrary characteristics. More precisely, we investigate dynamical properties of the conjugation action of such lattices on their space of positive definite functions. Our main results deal with the existence and the classification of characters from which we derive applications to topological dynamics, ergodic theory, unitary representations and operator algebras. Our key theorem is an extension of the noncommutative Nevo-Zimmer structure theorem obtained in [BH19] to the case of simple algebraic groups defined over arbitrary local fields. We also deduce a noncommutative analogue of Margulis' factor theorem for von Neumann subalgebras of the noncommutative Poisson boundary of higher rank arithmetic groups.

math.OA↗

arXiv2023

The noncommutative factor theorem for lattices in product groups

We prove a noncommutative Bader-Shalom factor theorem for lattices with dense projections in product groups. As an application of this result and our previous works, we obtain a noncommutative Margulis factor theorem for all irreducible lattices $Γ< G$ in higher rank semisimple algebraic groups. Namely, we give a complete description of all intermediate von Neumann subalgebras $\operatorname{L}(Γ) \subset M \subset \operatorname{L}(Γ\curvearrowright G/P)$ sitting between the group von Neumann algebra and the group measure space von Neumann algebra associated with the action on the Furstenberg-Poisson boundary.

math.OA↗

arXiv2022

Existentially closed W*-probability spaces

We study several model-theoretic aspects of W$^*$-probability spaces, that is, $σ$-finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W$^*$-spaces and prove several structural results about such spaces, including that they are type III$_1$ factors that tensorially absorb the Araki-Woods factor $R_\infty$. We also study the existentially closed objects in the restricted class of W$^*$-probability spaces with Kirchberg's QWEP property, proving that $R_\infty$ itself is such an existentially closed space in this class. Our results about existentially closed probability spaces imply that the class of type III$_1$ factors forms a $\forall_2$-axiomatizable class. We show that for $λ\in (0,1)$, the class of III$_λ$ factors is not $\forall_2$-axiomatizable but is $\forall_3$-axiomatizable; this latter result uses a version of Keisler's Sandwich theorem adapted to continuous logic. Finally, we discuss some results around elementary equivalence of III$_λ$ factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any $λ\in (0,1)$, there is a family of pairwise non-elementarily equivalent III$_λ$ factors of size continuum. While we cannot prove the same result for III$_1$ factors, we show that there are at least three pairwise non-elementarily equivalent III$_1$ factors by showing that the class of full factors is preserved under elementary equivalence.

math.OA↗

arXiv2022

Pointwise inner automorphisms of almost periodic factors

We prove that a large class of nonamenable almost periodic type ${\rm III_1}$ factors $M$, including all McDuff factors that tensorially absorb $R_\infty$ and all free Araki-Woods factors, satisfy Haagerup-Stormer's conjecture (1988): any pointwise inner automorphism of $M$ is the composition of an inner and a modular automorphism.

math.OA↗

arXiv2022

Charmenability of arithmetic groups of product type

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees.

math.GR↗

arXiv2021

Noncommutative ergodic theory of higher rank lattices

We survey recent results regarding the study of dynamical properties of the space of positive definite functions and characters of higher rank lattices. These results have several applications to ergodic theory, topological dynamics, unitary representation theory and operator algebras. The key novelty in our work is a dynamical dichotomy theorem for equivariant faithful normal unital completely positive maps between noncommutative von Neumann algebras and the space of bounded measurable functions defined on the Poisson boundary of semisimple Lie groups.

math.OA↗

arXiv2021

Stationary characters on lattices of semisimple Lie groups

We show that stationary characters on irreducible lattices $Γ< G$ of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice $Γ< G$, the left regular representation $λ_Γ$ is weakly contained in any weakly mixing representation $π$. We prove that for any such irreducible lattice $Γ< G$, any uniformly recurrent subgroup (URS) of $Γ$ is finite, answering a question of Glasner-Weiss. We also obtain a new proof of Peterson's character rigidity result for irreducible lattices $Γ< G$. The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

math.GR↗

arXiv2020

Examples of property (T) II$_1$ factors with trivial fundamental group

In this article we provide the first examples of property (T) $\rm II_1$ factors $\mathcal N$ with trivial fundamental group, $\mathcal F (\mathcal N)=1$. Our examples arise as group factors $\mathcal N=\mathcal L(G)$ where $G$ belong to two distinct families of property (T) groups previously studied in the literature: the groups introduced by Valette in \cite{Va04} and the ones introduced recently in \cite{CDK19} using the Belegradek-Osin Rips construction from \cite{BO06}. In particular, our results provide a continuum of explicit pairwise non-isomorphic property (T) factors.

math.OA↗

arXiv2020

Connes' bicentralizer problem for q-deformed Araki-Woods algebras

Let $(H_{\mathbf{R}}, U_t)$ be any strongly continuous orthogonal representation of $\mathbf{R}$ on a real (separable) Hilbert space $H_{\mathbf{R}}$. For any $q\in (-1,1)$, we denote by $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ the $q$-deformed Araki-Woods algebra introduced by Shlyakhtenko and Hiai. In this paper, we prove that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ has trivial bicentralizer if it is a type $\rm III_1$ factor. In particular, we obtain that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ always admits a maximal abelian subalgebra that is the range of a faithful normal conditional expectation. Moreover, using Sniady's work, we derive that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ is a full factor provided that the weakly mixing part of $(H_{\mathbf{R}}, U_t)$ is nonzero.

math.OA↗

arXiv2018

Structure of extensions of free Araki-Woods factors

We investigate the structure of crossed product von Neumann algebras arising from Bogoljubov actions of countable groups on Shlyakhtenko's free Araki-Woods factors. Among other results, we settle the questions of factoriality and Connes' type classification. We moreover provide general criteria regarding fullness and strong solidity. As an application of our main results, we obtain examples of type ${\rm III_0}$ factors that are prime, have no Cartan subalgebra and possess a maximal amenable abelian subalgebra. We also obtain a new class of strongly solid type ${\rm III}$ factors with prescribed Connes' invariants that are not isomorphic to any free Araki-Woods factors.

math.OA↗