Related papers: Connes' bicentralizer problem for q-deformed Araki…

In this article, we show that the mixed $q$-deformed Araki-Woods von Neumann algebra $\Gamma_T(H_\R, U_t)^{\prime\prime}$ has trivial bicentralizer, whenever it is of type $\mathrm{III}_1$.

The $q$-deformed Araki-Woods von Neumann algebras $\Gamma_q(\mathcal{H}_\mathbb{R}, U_t)^{\prime \prime}$ 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…

We prove that the mixed $q$-Gaussian algebra $\Gamma_{Q}(H_{\mathbb{R}})$ 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 show that for any free Araki-Woods factor $\mathcal{M} = \Gamma(H_\R, U_t)"$ of type ${\rm III_1}$, the bicentralizer of the free quasi-free state $\varphi_U$ is trivial. Using Haagerup's Theorem, it follows that there always exists a…

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…

It is proved that the $q$-Araki-Woods von Neumann algebras $\Gamma_q(\CH_\R,U_t)^{\prime\prime}$ for $q\in (-1,1)$ are factors if $dim(\CH_\R)\geq 3$.

We show that for any type ${\rm III_1}$ free Araki-Woods factor $\mathcal{M} = \Gamma(H_\R, U_t)"$ associated with an orthogonal representation $(U_t)$ of $\R$ on a separable real Hilbert space $H_\R$, the continuous core $M = \mathcal{M}…

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 $\Gamma(H_{\mathbb R},…

We show that all the free Araki-Woods factors $\Gamma(H_\R, U_t)"$ have the complete metric approximation property. Using Ozawa-Popa's techniques, we then prove that every nonamenable subfactor $\mathcal{N} \subset \Gamma(H_\R, U_t)"$ which…

We define a new $q$-deformation of Brauer's centralizer algebra which contains Hecke algebras of type $A$ as unital subalgebras. We determine its generic structure as well as the structure of certain semisimple quotients. This is expected…

It is proved that the $q$-Araki-Woods factor $\Gamma_q(\sH_\R, U)''$ associated with a strongly continuous orthogonal representation $U:\R\to \cO(\sH_\R)$ is strongly solid for all $q\in (-1,1)$ if the representation $U$ is almost periodic.…

We establish factoriality and non-injectivity in full generality for the mixed $q$-Araki-Woods von Neumann algebra associated to a separable real Hilbert space $\mathsf{H}_{\mathbf{R}}$ with $\dim\mathsf{H}_{\mathbf{R}}\geq 2$, a strongly…

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D4,E6,E7,E8). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to…

Inside the double affine Hecke algebra of type $GL_n$, which depends on two parameters $q$ and $\tau$, we define a subalgebra $\mathbb{H}^{\mathfrak{gl}_n}$ that may be thought of as a $q$-analogue of the degree zero part of the…

We give a new proof of a theorem due to Alain Connes, that an injective factor $N$ of type III$_1$ with separable predual and with trivial bicentralizer is isomorphic to the Araki--Woods type III$_1$ factor $R_{\infty}$. This, combined with…

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…

Clifford geometric algebras of multivectors are introduced which exhibit a bilinear form which is not necessarily symmetric. Looking at a subset of bi-vectors in CL(K^{2n},B), we proof that theses elements generate the Hecke algebra…

In this article, we give a classification of irreducible completely splittable representations of affine Hecke-Clifford superalgebras $H_n^{\mathrm{aff}}(q)$ when $q^2$ is a primitive $h$-th root of unity. As an application, we derive a…

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…

The $q$-Araki-Woods factor associated to a group of orthogonal transformations on a real separable Hilbert space $\mathsf{H}_{\mathbb{R}}$ is full as soon as $\dim\mathsf{H}_{\mathbb{R}}\geq 2$.