Related papers: On the uniqueness of injective III$_1$ factor

We give an alternative proof that an injective factor on a Hilbert space with trivial bicentralizer is ITPFI. Our proof is given in parallel with each type of factors and it is based on the strategy of Haagerup. As a consequence, the…

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…

We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic…

We make a series of model-theoretic contributions to Connes' bicentralizer problem, one of the most prominent open problems in the theory of von Neumann algebras. Our work builds on the recent result of Houdayer and Marrakchi who show that,…

We show the relative bicentralizer flow and the relative flow of weights are isomorphic for an inclusion of injective factors of type III$_1$ with finite index, or an irreducible discrete inclusion whose small algebra is an injective factor…

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 \tau cannot distinguish all isomorphism…

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 $\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ the…

The canonical modular homomorphism provides an embedding of $\mathbb{R}$ into the outer automorphism group Out($M$) of any type III$_{1}$ factor $M$. We provide an explicit construction of a full factor of type III$_{1}$ with separable…

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…

We show that any type ${\rm III_1}$ factor with separable predual satisfying Connes' Bicentralizer Property (CBP) has a singular maximal abelian $\ast$-subalgebra that is the range of a normal conditional expectation. We also investigate…

Let $R_\infty$ denote the Araki--Woods factor -- the unique separable injective type III$_{1}$ factor. For extremal almost periodic states $\varphi, \psi\in (R_\infty)_*$, we show that if $\Delta_\varphi$ and $\Delta_\psi$ have the same…

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 $\mathrm{III}_1$.

We show that a factor $M$ is full if and only if the $C^*$-algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of a type $\mathrm{III}_1$ factor is always…

This paper addresses a conjecture of Kadison and Kastler that a von Neumann algebra M on a Hilbert space H should be unitarily equivalent to each sufficiently close von Neumann algebra N and, moreover, the implementing unitary can be chosen…

We show that any almost periodic outer flow $\alpha : \mathbb R \curvearrowright R$ on the hyperfinite type $\mathrm{II}_1$ factor with Connes' spectrum $\Gamma(\alpha) = \mathbb R$ satisfies the Rokhlin property and thus is unique up to…

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}…

In this paper we consider the following problem: When are the preduals of two hyperfinite (=injective) factors $\M$ and $\N$ (on separable Hilbert spaces) cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if $\M$ is…

By developing a theory of anticoarse spaces in the purely infinite setting and using 1-bounded entropy techniques along with recent strong convergence results in random matrix theory, we show that free Araki--Woods factors offer the first…

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…

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'…