Panchugopal Bikram, Kunal Mukherjee, Éric Ricard, Simeng Wang
The $q$-deformed Araki-Woods von Neumann algebras $Γ_q(\mathcal{H}_\mathbb{R}, U_t)^{\prime \prime}$ are factors for all $q\in (-1,1)$ whenever $dim(\mathcal{H}_\mathbb{R})\geq 3$. When $dim(\mathcal{H}_\mathbb{R})=2$ they are factors as well for all $q$ so long as the parameter defining $(U_t)$ is `small' or $1$ $($trivial$)$ as the case may be.
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It is proved that the $q$-Araki-Woods von Neumann algebras $Γ_q(\CH_\R,U_t)^{\prime\prime}$ for $q\in (-1,1)$ are factors if $dim(\CH_\R)\geq 3$.
To any strongly continuous orthogonal representation of $\R$ on a real Hilbert space $\CH_\R$, Hiai constructed $q$-deformed Araki-Woods von Neumann algebras for $-1< q< 1$, which are $W^{\ast}$-algebras arising from non tracial representations of the $q$-commutation relations, the latter yielding an interpolation between the Bosonic and Fermionic statistics. We prove that if the orthogonal representation is not ergodic then these von Neumann algebras are factors whenever $dim(\CH_\R)\geq 2$ and $q\in (-1,1)$. In such case, the centralizer of the $q$-quasi free state has trivial relative commu…
Full solution of the factoriality question for $q$-Araki-Woods von Neumann algebras via conjugate variables
利用共轭变量方法及抽象结果，完全证明了一般情形下生成元数至少为二的 $q$-Araki-Woods 冯·诺伊曼代数的因子性，并确立其非内射性、确定因子类型及有限生成时的固体性与完全性。
Using Speicher central limit Theorem we provide Hiai's q-Araki-Woods von Neumann algebras with nice asymptotic matricial models. Then, we use this model and an elaborated ultraproduct procedure, to show that all q-Araki-Woods von Neumann algebras are QWEP.
We show that Ozawa's recent results on solid von Neumann algebras imply that there are free Araki-Woods factors, which fail to have free absorption. We also show that a free Araki-Woods factors $Γ(μ, n)$ associated to a measure and a multiplicity function $n$ may non-trivially depend on the multiplicity function.
We extend the class of contractions for which the second quantisation on $q$-Araki-Woods algebras can be defined. As a corollary, we prove that all $q$-Araki-Woods algebras possess the Haagerup approximation property.
The generating abelian subalgebras arsing from vectors in the ergodic component of Hiai's construction of the q-deformed Araki-Woods von Neumann algebras are quasi-split.
We show that any amenable von Neumann subalgebra of any free Araki-Woods factor that is globally invariant under the modular automorphism group of the free quasi-free state is necessarily contained in the almost periodic free summand.
Complete Metric Approximation Property For Mixed $q$-Deformed Araki-Woods Factors
该研究利用超积嵌入方法，针对元素在-1到1之间的对称矩阵，证明了混合$q$-形变Araki-Woods因子具有弱*完全度量逼近性质，丰富了算子代数领域的相关理论。
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.
Conjugate variables approach to mixed $q$-Araki-Woods algebras: Factoriality and non-injectivity
本文利用共轭变量方法，在最一般情形下证明了与可分实Hilbert空间等结构关联的混合$q$-Araki-Woods von Neumann代数的因子性与非内射性。
Connes' Bicentralizer Problem for Mixed $q$-deformed Araki-Woods Algebras
本文证明了型 $\mathrm{III}_1$ 的混合 $q$-变形 Araki-Woods von Neumann 代数具有平凡的双中心化子，解决了 Connes 双中心化子问题。
Structural and non-isomorphism results for $q$-Araki-Woods factors
本文证明几乎周期表示对应的 $q$-Araki-Woods 因子是强固体的，并指出在特定条件下它不与自由 Araki-Woods 因子同构。
We show that any free product of finite-dimensional von Neumann algebras equipped with non-tracial states is isomorphic to a free Araki-Woods factor with its free quasi-free state possibly direct sum a finite-dimensional von Neumann algebra. This gives a complete answer to questions posed by Dykema and Shlyakhtenko, which had been partially answered by work of Houdayer and work of Ueda. We also extend this to suitable infinite-dimensional von Neumann algebras with almost periodic states.
We prove that the mixed $q$-Gaussian algebra $Γ_{Q}(H_{\mathbb{R}})$ associated to a real Hilbert space $H_{\mathbb{R}}$ and a real symmetric matrix $Q=(q_{ij})$ with $\sup|q_{ij}|<1$, is a factor as soon as $\dim H_{\mathbb{R}}\geq2$. We also discuss the factoriality of $q$-deformed Araki-Woods algebras, in particular showing that the $q$-deformed Araki-Woods algebra $Γ_{q}(H_{\mathbb{R}},U_{t})$ given by a real Hilbert space $H_{\mathbb{R}}$ and a strongly continuous group $U_{t}$ is a factor when $\dim H_{\mathbb{R}}\geq2$ and $U_{t}$ admits an invariant eigenvector.
Free Araki-Woods factors and Connes' bicentralizer problem
本文证明了型III_1自由Araki-Woods因子的自由拟自由态双中心化子平凡，并利用Haagerup定理得出存在满足特定交换条件的忠实正规态，从而推进了Connes双中心化子问题的研究。
Structure of extensions of free Araki-Woods factors
本文研究了可数群对自由Araki-Woods因子的Bogoljubov作用产生的交叉积von Neumann代数结构，解决了因子性与Connes型分类问题，给出了丰满性与强坚固性准则，并构造出具有特定性质的III型因子新例子。
On some free products of von Neumann algebras which are free Araki-Woods factors
本文证明了I型因子等von Neumann代数在非迹几乎周期态下的自由积是自由Araki-Woods因子，其完全分类由Connes的Sd不变量给出，并回答了Dykema和Shlyakhtenko的相关问题。
By adapting an ultraproduct technique of Junge and Zeng, we prove that radial completely bounded multipliers on $q$-Gaussian algebras transfer to $q$-Araki-Woods algebras. As a consequence, we establish the $w^{\ast}$-complete metric approximation property for all $q$-Araki-Woods algebras. We apply the latter result to show that the canonical ultraweakly dense C$^\ast$-subalgebras of $q$-Araki-Woods algebras are always QWEP.
Classification of a family of non almost periodic free Araki-Woods factors
本文通过证明自由 Araki-Woods 因子的联合测度类不变量，并建立忠实正规态酉共轭的形变/刚性判据，完成了一大类非几乎周期自由 Araki-Woods 因子的完全分类，并由此得到自由积 von Neumann 代数的刚性与分类定理。
We introduce a new invariant \mathcal{S}(M) for type III factors M with no almost-periodic weights. We compute this invariant for certain free Araki-Woods factors. We show that Connes' invariant τcannot distinguish all isomorphism classes of free Araki-Woods factors. We show that there exists a continuum of mutually non-isomorphic free Araki-Woods factors, each without almost-periodic weights.
We consider semi-group BMO-spaces associated with arbitrary von Neumann algebras and prove interpolation theorems. This extends results by Junge-Mei for the tracial case. We give examples of multipliers on free Araki-Woods algebras and in particular we find $L_\infty \rightarrow {\rm BMO}$ multipliers. We also provide $L_p$-bounds for a natural generalization of the Hilbert transform.
Fullness of $q$-Araki-Woods factors
本文证明了与实可分Hilbert空间上正交变换群相关联的$q$-Araki-Woods因子，在空间维度不小于2时即具有丰满性。
Using Baire category techniques we prove that Araki-Woods factors are not classifiable by countable structures. As a result, we obtain a far reaching strengthening as well as a new proof of the well-known theorem of Woods that the isomorphism problem for ITPFI factors is not smooth. We derive as a consequence that the odometer actions of Z that preserve the measure class of a finite non-atomic product measure are not classifiable up to orbit equivalence by countable structures.
Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the $q$-Ornstein-Uhlenbeck semigroup on the $q$-deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the $q$-Araki-Woods algebras. We show that hypercontractivity from $L^p$ to $L^2$ can occur if and only if the generator of the deformation is bounded.
We define the standard Borel space of free Araki-Woods factors and prove that their isomorphism relation is not classifiable by countable structures. We also prove that equality of $τ$-topologies, arising as invariants of type III factors, as well as coycle and outer conjugacy of actions of abelian groups on free product factors are not classifiable by countable structures.
Asymptotic structure of free Araki-Woods factors
本文利用超积 von Neumann 代数框架研究自由 Araki-Woods 因子的结构，证明了其 $ω$-刚性、连续核的二分法等渐近性质。
Structural results for free Araki-Woods factors and their continuous cores
本文证明了自由 Araki-Woods 因子的连续核在一般情形下是半固的，在表示为混合时是固的，并构造了一个具有完整基本群且非同构于插值自由群因子的非顺从固因子实例。
Twisted Araki-Woods Algebras, the Yang-Baxter Equation, and quantum field theory
本文回顾了twisted Araki-Woods代数的研究，介绍了twisted Fock空间，并探讨了源自集合论Yang-Baxter方程的新示例族及其在构建量子场论模型中的相关性。
This is a survey paper on the work of D. Shlyakhtenko on free Araki-Woods factors. The paper was written (in French) on the occasion of the Bourbaki seminar. Our original contribution consists in computing the tau invariant for arbitrary free Araki-Woods factors.
We show that the operator Hilbert space OH introduced by Pisier embeds into the predual of the hyerfinite III1 factor. The main new tool is a Khintchine type inequality for the generators of the CAR algebra with respect to a quasi-free state. Our approach yields a Khintchine type inequality for the q-gaussian variables for all values q between -1 and 1. These results are closely related to recent results of Pisier and Shlyakhtenko in the free case.
Operator-Valued Twisted Araki-Woods Algebras
本文引入算子值扭曲Araki-Woods代数，发展将其归结为标量值情形的解体理论，刻画其自然权重的模理论，并给出保证代数因子性的充分判据。
Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors
本文证明自由 Araki-Woods 因子具有完全度量逼近性质，并利用 Ozawa-Popa 技术证明特定非顺从子因子无 Cartan 子代数，最终解决了 Connes 构造的 III_1 型因子与自由 Araki-Woods 因子不同构的问题。
A Conjugate System for Twisted Araki-Woods von Neumann Algebras of finite dimensional spaces
本文计算了有限维空间上满足特定范数条件的扭曲 Araki-Woods von Neumann 代数的共轭系统，证明其具有有限的非微态自由 Fisher 信息并为特定型因子，且在范数足够小时同构于自由 Araki-Woods 代数。
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 e…
Modular Structure and Inclusions of Twisted Araki-Woods Algebras
本文分析了扭曲 Araki-Woods 代数的模结构与包含关系，证明了 Fock 真空循环分离的条件并显式确定了 Tomita-Takesaki 模数据，同时探讨了包含关系的奇异性与 III 型相对交换子，最后应用于代数量子场论中的可观测量的定域化。
We show that for each (0<λ<1), the free Araki-Woods factor of type III(_λ) cannot be written as a tensor product of two diffuse von Neumann algebras (i.e., is prime), and does not contain a Cartan subalgebra.
We show that for F an invertible 2 by 2 matrix, the von Neumann algebra associated to the universal quantum group A_u(F) is a free Araki-Woods factor.
Unique prime factorization and bicentralizer problem for a class of type III factors
针对包含自由 Araki-Woods 因子的大类von Neumann代数，证明了张量积的唯一素分解及Connes双中心化子问题的正解。
We construct interacting quantum fields in 1+1 space-time dimensions, representing charged or neutral scalar bosons at positive temperature and zero chemical potential. Our work is based on prior work by Klein and Landau and Hoegh-Krohn. Generalized path space methods are used to add a spatially cut-off interaction to the free system, which is described in the Araki-Woods representation. It is shown that the interacting KMS state is normal w.r.t. the Araki-Woods representation. The observable algebra and the modular conjugation of the interacting system are shown to be identical to the ones of…
We prove that the von Neumann algebra generated by q-gaussians is not injective as soon as the dimension of the underlying Hilbert space is greater than 1. Our approach is based on a vector valued Khintchine type inequality for Wick products. The same proof also works for the more general setting of a Yang-Baxter deformation. Our techniques can also be extended to the so called q-Araki-Woods von Neumann algebras recently introduced by Hiai. In this latter case, we obtain the non injectivity under some asssumption on the spectral set of the positive operator associated with the deformation.
Fock structure of complete Boolean algebras of type I factors and of unital factorizations
本文证明I型因子完备布尔代数的可分解向量具有完全性，解决Araki-Woods猜想，并用含单位因子分解的非交换语言刻画Fock型结构。
Given a finite, directed, connected graph $Γ$ equipped with a weighting $μ$ on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional $(\mathcal{M}(Γ,μ),φ)$. When the weighting $μ$ is instead on the vertices of $Γ$, the first author showed the isomorphism class of $(\mathcal{M}(Γ,μ),φ)$ depends only on the data $(Γ,μ)$ and is an interpolated free group factor equipped with a scaling of its unique trace (possibly direct sum copies of $\mathbb{C}$). Moreover, the free dimension of the interpolated free group factor is easily com…
Coarse rigidity in von Neumann algebras via derivations
本文通过模导子提出冯·诺依曼代数的相对粗刚性概念，扩展L²-刚性至非迹情形，并证明无相对amenable项时形变收敛，从而统一证明完全性等不可分解性质。
On von Neumann algebras generated by free Poisson random weights
本文通过引入 n.s.f. 权重推广了自由泊松随机测度，构造了自由泊松冯·诺依曼代数，研究了其分解与因子条件，并揭示了自由 Araki-Woods 代数与自由 Lévy 过程的内在联系。
We investigate the singular subspace of an inclusion of tracial von Neumann algebras. The singular subspace is a canonical N-N subbimodule of L^{2}(M) and it contains the quasinormalizer introduced by Popa, one-sided quasinormalizer introduced by Fang-Gao-Smith, and wq-normalizer introduced in Galatan-Popa (following upon work in Ioana-Peterson-Popa and Popa). We then obtain a weak notion of regularity (called spectral regularity) by demanding that the singular subspace of N in M generates M. By abstracting Voiculescu's original proof of absence of Cartan subalgebras, we show that there can be…
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