arXiv subjects

Cyril Houdayer

Publications and source records attributed to Cyril Houdayer.

51 records · Page 3Linked to original sources

arXiv2014

Structure of ${\rm II_1}$ factors arising from free Bogoljubov actions of arbitrary groups

In this paper, we investigate several structural properties for crossed product ${\rm II_1}$ factors $M$ arising from free Bogoljubov actions associated with orthogonal representations $π: G \to \mathcal O(H_\mathbf R)$ of arbitrary countable discrete groups. Under fairly general assumptions on the orthogonal representation $π: G \to \mathcal O(H_{\mathbf R})$, we show that $M$ does not have property Gamma of Murray and von Neumann. Then we show that any regular amenable subalgebra $A \subset M$ can be embedded into $L(G)$ inside $M$. Finally, when $G$ is assumed to be amenable, we locate precisely any possible amenable or Gamma extension of $L(G)$ inside $M$.

math.OA↗

arXiv2014

A class of ${\rm II_1}$ factors with an exotic abelian maximal amenable subalgebra

We show that for every mixing orthogonal representation $π: \Z \to \mathcal O(H_\R)$, the abelian subalgebra $\LL(\Z)$ is maximal amenable in the crossed product ${\rm II}_1$ factor $Γ(H_\R)\dpr \rtimes_π\Z$ 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 $\LL^2(M \ominus A)$ where $A \subset M$ is a maximal amenable masa in a ${\rm II_1}$ factor.

math.OA↗

arXiv2014

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

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ 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 $(M_1, φ_1) \ast (M_2, φ_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined on a standard measure space and which splits as the free product $\mathcal R = \mathcal R_1 \ast \mathcal R_2$ of recurrent subequivalence relations gives rise to a nonamenable factor $\rL(\mathcal R)$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors $\rL^\infty(X) \rtimes Γ$ arising from nonsingular free ergodic actions $Γ\curvearrowright (X, μ)$ on standard measure spaces of amalgamated groups $Γ= Γ_1 \ast_Σ Γ_2$ over a finite subgroup $Σ$.

math.OA↗

arXiv2012

Type III factors with unique Cartan decomposition

We prove that for any free ergodic nonsingular nonamenable action Γ \actson (X,μ) of all Γ in a large class of groups including all hyperbolic groups, the associated group measure space factor $L^\infty(X) \rtimes Γ$ has L^\infty(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 $M_1 *_B M_2$ over a subalgebra B of type I.

math.OA↗

arXiv2011

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 Γcovering certain cases where Γis an amalgamated free product over a non-amenable subgroup. In combination with Kida's work we deduce that if Σ< SL(3,\Z) denotes the subgroup of matrices g with g_31 = g_32 = 0, then any free ergodic probability measure preserving action of Γ= SL(3,\Z) *_ΣSL(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.

math.OA↗

arXiv2011

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

We show that all the free Araki-Woods factors $Γ(H_\R, U_t)"$ have the complete metric approximation property. Using Ozawa-Popa's techniques, we then prove that every nonamenable subfactor $\mathcal{N} \subset Γ(H_\R, U_t)"$ which is the range of a normal conditional expectation has no Cartan subalgebra. We finally deduce that the type ${\rm III_1}$ factors constructed by Connes in the '70s can never be isomorphic to any free Araki-Woods factor, which answers a question of Shlyakhtenko and Vaes.

math.OA↗

arXiv2011

Invariant percolation and measured theory of nonamenable groups

Using percolation techniques, Gaboriau and Lyons recently proved that every countable, discrete, nonamenable group $Γ$ contains measurably the free group $\mathbf F_2$ on two generators: there exists a probability measure-preserving, essentially free, ergodic action of $\mathbf F_2$ on $([0, 1]^Γ, λ^Γ)$ such that almost every $Γ$-orbit of the Bernoulli shift splits into $\mathbf F_2$-orbits. A combination of this result and works of Ioana and Epstein shows that every countable, discrete, nonamenable group admits uncountably many non-orbit equivalent actions.

math.GR↗

arXiv2011

Strongly solid ${\rm II_1}$ factors with an exotic MASA

Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid $\rm{II}_1$ factor $M$ containing an "exotic" maximal abelian subalgebra $A$: as an $A$,$A$-bimodule, $L^2(M)$ is neither coarse nor discrete. Thus we show that there exist $\rm{II}_1$ factors with such property but without Cartan subalgebras. It also follows from Voiculescu's free entropy results that $M$ is not an interpolated free group factor, yet it is strongly solid and has both the Haagerup property and the complete metric approximation property.

math.OA↗

arXiv2010

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

We show that for any type ${\rm III_1}$ free Araki-Woods factor $\mathcal{M} = Γ(H_\R, U_t)"$ associated with an orthogonal representation $(U_t)$ of $\R$ on a separable real Hilbert space $H_\R$, the continuous core $M = \mathcal{M} \rtimes_σ\R$ is a semisolid ${\rm II_\infty}$ factor, i.e. for any non-zero finite projection $q \in M$, the ${\rm II_1}$ factor $qMq$ is semisolid. If the representation $(U_t)$ is moreover assumed to be mixing, then we prove that the core $M$ is solid. As an application, we construct an example of a non-amenable solid ${\rm II_1}$ factor $N$ with full fundamental group, i.e. $\mathcal{F}(N) = \R^*_+$, which is not isomorphic to any interpolated free group factor $L(\F_t)$, for $1 < t \leq +\infty$.

math.OA↗

arXiv2010

Bass-Serre rigidity results in von Neumann algebras

We obtain new Bass-Serre type rigidity results for ${\rm II_1}$ equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any non-amenable factor arising as an amalgamated free product of von Neumann algebras $\mathcal{M}_1 \ast_B \mathcal{M}_2$ over an abelian von Neumann algebra $B$, is prime, i.e. cannot be written as a tensor product of diffuse factors. This gives, both in the type ${\rm II_1}$ and in the type ${\rm III}$ case, new examples of prime factors.

math.OA↗

arXiv2010

Strongly solid group factors which are not interpolated free group factors

We give examples of non-amenable ICC groups $Γ$ with the Haagerup property, weakly amenable with constant $Λ_{\cb}(Γ) = 1$, for which we show that the associated ${\rm II_1}$ factors $L(Γ)$ are strongly solid, i.e. the normalizer of any diffuse amenable subalgebra $P \subset L(Γ)$ generates an amenable von Neumann algebra. Nevertheless, for these examples of groups $Γ$, $L(Γ)$ is not isomorphic to any interpolated free group factor $L(\F_t)$, for $1 < t \leq \infty$.

math.OA↗

arXiv2009

Construction of type ${\rm II_1}$ factors with prescribed countable fundamental group

In the context of Free Probability Theory, we study two different constructions that provide new examples of factors of type ${\rm II_1}$ with prescribed fundamental group. First we investigate state-preserving group actions on the almost periodic free Araki-Woods factors satisfying both a condition of mixing and a condition of free malleability in the sense of Popa. Typical examples are given by the free Bogoliubov shifts. Take an ICC $w$-rigid group $G$ such that $\mathcal{F}(L(G)) = \{1\}$ (e.g. $G = \Z^2 \rtimes \SL(2, \Z)$). For any countable subgroup $S \subset \R^*_+$, we show that there exists an action of $G$ on $L(\F_\infty)$ such that $L(\F_\infty) \rtimes G$ is a type ${\rm II_1}$ factor and its fundamental group is $S$. The second construction is based on a free product. Take $(B(H), ψ)$ any factor of type ${\rm I}$ endowed with a faithful normal state and denote by $Γ\subset \R^*_+$ the subgroup generated by the point spectrum of $ψ$. We show that the centralizer $(L(G) \ast B(H))^{τ\ast ψ}$ is a type ${\rm II_1}$ factor and its fundamental group is $Γ$. Our proofs rely on Popa's deformation/rigidity strategy using his intertwining-by-bimodules technique.

math.OA↗

arXiv2009

Free Araki-Woods factors and Connes' bicentralizer problem

We show that for any free Araki-Woods factor $\mathcal{M} = Γ(H_\R, U_t)"$ of type ${\rm III_1}$, the bicentralizer of the free quasi-free state $φ_U$ is trivial. Using Haagerup's Theorem, it follows that there always exists a faithful normal state $ψ$ on $\mathcal{M}$ such that $(\mathcal{M}^ψ)' \cap \mathcal{M} = \C$.

math.OA↗

arXiv2008

On some free products of von Neumann algebras which are free Araki-Woods factors

We prove that certain free products of factors of type ${\rm I}$ and other von Neumann algebras with respect to nontracial, almost periodic states are almost periodic free Araki-Woods factors. In particular, they have the free absorption property and Connes' Sd invariant completely classifies these free products. For example, for $λ, μ\in ]0, 1[$, we show that $$(M_2(\C), ω_λ) \ast (M_2(\C), ω_μ)$$ is isomorphic to the free Araki-Woods factor whose Sd invariant is the subgroup of $\R^*_+$ generated by $λ$ and $μ$. Our proofs are based on algebraic techniques and amalgamated free products. These results give some answers to questions of Dykema and Shlyakhtenko.

math.OA↗

arXiv2006

A new construction of factors of type ${\rm III_1}$

We give in this paper a new construction of factors of type ${\rm III_1}$. Under certain assumptions, we can, thanks to a result by Popa, give a complete classification for this family of factors. Although these factors are never full, we can nevertheless, in many cases, compute Connes' $τ$ invariant. We obtain a new example of an uncountable family of pairwise non-isomorphic factors of type ${\rm III_1}$ with the same $τ$ invariant.

math.OA↗