arXiv subjects

Yusuke Isono

Publications and source records attributed to Yusuke Isono.

At least 19 recordsLinked to original sources

arXiv2025

Cocycle perturbations and ergodicity for actions on type III factors

We study cocycle perturbations of state preserving actions on type $\mathrm{III}_1$ factors. Extending the theorem of Marrakchi and Vaes for type $\mathrm{II}_1$ factors, we show that a state preserving outer $\mathbb Z$-action on a type $\mathrm{III}_1$ factor with trivial bicentralizer admits a unitary cocycle whose perturbation becomes an ergodic action. This partially answers a question of Marrakchi and Vaes. A major difference from the type $\mathrm{II}$ case is that the modular automorphism group naturally appears as part of the action, making the construction of the required cocycle more delicate.

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arXiv2025

Weak relative Dixmier property and Popa's intertwining technique for type III subfactors

Let $ A \subset M $ be an inclusion of von Neumann algebras equipped with a faithful normal semifinite operator valued weight $ E \colon M \to A $. We prove that every positive element $ x \in M $ with $ E(x) < \infty $ satisfies the weak Dixmier property relative to $ A $: the $ \sigma $-weak closure of the convex hull of its unitary orbit under $ \mathcal{U}(A) $ intersects the relative commutant $ A' \cap M $. This extends Marrakchi's result for the case of conditional expectations. We apply this result to obtain new structural theorems for type III factors, including a reformulation of Popa's intertwining criterion without tracial assumptions, an extension of Ozawa's relative solidity theorem to the type III setting, and a Galois-type correspondence for crossed products by totally disconnected groups. The last result resolves a question posed by Boutonnet and Brothier regarding the structure of intermediate subfactors.

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arXiv2023

Haagerup and St{\o}rmer's conjecture for pointwise inner automorphisms

In 1988, Haagerup and St{\o}rmer conjectured that any pointwise inner automorphism of a type $\rm III_1$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that any type $\rm III_1$ factor with trivial bicentralizer indeed satisfies this condition. In particular, this shows that Haagerup and St{\o}rmer's conjecture holds in full generality if Connes' bicentralizer problem has an affirmative answer. Our proof is based on Popa's intertwining theory and Marrakchi's recent work on relative bicentralizers.

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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.

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arXiv2021

Note on bi-exactness for creation operators on Fock spaces

In this note, we introduce and study a notion of bi-exactness for creation operators acting on full, symmetric and anti-symmetric Fock spaces. This is a generalization of our previous work, in which we studied the case of anti-symmetric Fock spaces. As a result, we obtain new examples of solid actions as well as new proofs for some known solid actions. We also study free wreath product groups in the same context.

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arXiv2020

Ergodic theory of affine isometric actions on Hilbert spaces

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type $\mathrm{III}_1$.

math.DS↗

arXiv2020

Boundary and rigidity of nonsingular Bernoulli actions

Let $ G $ be a countable discrete group and consider a nonsingular Bernoulli shift action $ G \curvearrowright \prod_{g\in G }(\{0,1\},μ_g)$ with two base points. When $ G $ is exact, under a certain finiteness assumption on the measures $\{μ_g\}_{g\in G }$, we construct a boundary for the Bernoulli crossed product C$^*$-algebra that admits some commutativity and amenability in the sense of Ozawa's bi-exactness. As a consequence, we obtain that any such Bernoulli action is solid. This generalizes solidity of measure preserving Bernoulli actions by Ozawa and Chifan--Ioana, and is the first rigidity result in the non measure preserving case. For the proof, we use anti-symmetric Fock spaces and left creation operators to construct the boundary and therefore the assumption of having two base points is crucial.

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arXiv2020

$L_2$-cohomology, derivations and quantum Markov semi-groups on $q$-Gaussian algebras

We study (quasi-)cohomological properties through an analysis of quantum Markov semi-groups. We construct higher order Hochschild cocycles using gradient forms associated with a quantum Markov semi-group. By using Schatten-$\mathcal{S}_p$ estimates we analyze when these cocycles take values in the coarse bimodule. For the 1-cocycles (the derivations) we show that under natural conditions they imply the Akemann-Ostrand property (using the Riesz transform). We apply this to $q$-Gaussian algebras $Γ_q(H)$. As a result $q$-Gaussians satisfy AO$^+$ for $| q | \leqslant \dim(H)^{-1/2}$. This includes a new range of $q$ in low dimensions compared to Shlyakhtenko.

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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.

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arXiv2019

Tensor product decompositions and rigidity of full factors

We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products coming from compact actions of higher rank lattices (e.g.\ $\mathrm{SL}(n,\mathbb{Z}), \: n \geq 3$) and noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable). Finally, we provide examples of full factors without any prime factorization.

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arXiv2019

Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

Let $\mathbb{G}$ be a free (unitary or orthogonal) quantum group. We prove that for any non-amenable subfactor $N\subset L^\infty(\mathbb{G})$, which is an image of a faithful normal conditional expectation, and for any $σ$-finite factor $B$, the tensor product $N \otimes B$ has no Cartan subalgebras. This generalizes our previous work that provides the same result when $B$ is finite. In the proof, we establish Ozawa--Popa and Popa--Vaes's weakly compact action on the continuous core of $N \otimes B$ as the one relative to B, by using an operator valued weight to B and the central weak amenability of $\mathbb{G}$.

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arXiv2019

On fundamental groups of tensor product $\rm II_1$ factors

Let $M$ be a $\rm II_1$ factor and let $\mathcal{F}(M)$ denote the fundamental group of $M$. In this article, we study the following property of $M$: for arbitrary $\rm II_1$ factor $B$, we have $\mathcal{F}(M \overline{\otimes} B)=\mathcal{F}(M)\mathcal{F}(B)$. We prove that for any subgroup $G\leq \mathbb{R}^*_+$ which is realized as a fundamental group of a $\rm II_1$ factor, there exists a $\rm II_1$ factor $M$ which satisfies this property and whose fundamental group is $G$. Using this, we deduce that if $G,H \leq \mathbb{R}^*_+$ are realized as fundamental groups of $\rm II_1$ factors (with separable predual), then so are groups $G \cdot H$ and $G \cap H$.

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arXiv2019

Unique prime factorization for infinite tensor product factors

In this article, we investigate a unique prime factorization property for infinite tensor product factors. We provide several examples of type II and III factors which satisfy this property, including all free product factors with diffuse free product components. In the type III setting, this is the first classification result for infinite tensor product non-amenable factors. Our proof is based on Popa's intertwining techniques and a characterization of relative amenability on the continuous cores.

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arXiv2019

Unitary conjugacy for type III subfactors and W$^*$-superrigidity

Let $A,B\subset M$ be inclusions of $σ$-finite von Neumann algebras such that $A$ and $B$ are images of faithful normal conditional expectations. In this article, we investigate Popa's intertwining condition $A\preceq_MB$ using their modular actions. In the main theorem, we prove that if $A\preceq_MB$ holds, then an intertwining element for $A\preceq_MB$ also intertwines some modular flows of $A$ and $B$. As a result, we deduce a new characterization of $A\preceq_MB$ in terms of their continuous cores. Using this new characterization, we prove the first W$^*$-superrigidity type result for group actions on amenable factors. As another application, we characterize stable strong solidity for free product factors in terms of their free product components.

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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.

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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.

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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

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 $\mathcal C_{(\text{AO})}$.

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