Related papers: Model theory and Connes' bicentralizer problem

We introduce the notion of selfless W$^*$-probability space and study its connection with Connes' bicentralizer problem. In particular, we show that if $M$ is a separable type ${\rm III_1}$ factor with trivial bicentralizer, then $(M,…

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…

In 1967, Kadison asked "if $N$ is a subfactor of the factor $M$ for which $N' \cap M$ consists of scalars, will some maximal abelian *-subalgebra of $N$ be a maximal abelian subalgebra of $M$?". Generalizing a theorem of Popa in the type…

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 study several model-theoretic aspects of W$^*$-probability spaces, that is, $\sigma$-finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W$^*$-spaces and prove several structural…

We investigate the structure of the relative bicentralizer algebra ${\rm B}(N \subset M, \varphi)$ for inclusions of von Neumann algebras with normal expectation where $N$ is a type ${\rm III_1}$ subfactor and $\varphi \in N_*$ is a…

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…

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 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…

Let $G = G_{1} \times G_{2}$ be a product of two locally compact, second countable groups and $\mu \in \mathrm{Prob}(G)$ be of the form $\mu = \mu_{1} \times \mu_{2}$, where $\mu_{i} \in \mathrm{Prob}(G_{i})$. Let $(B,\nu_B)$ be the…

We introduce pseudofinite W*-probability spaces. These are W*-probability spaces that are elementarily equivalent to Ocneanu ultraproducts of finite-dimensional von Neumann algebras equipped with arbitrary faithful normal states. We are…

We show that Connes' embedding problem for II_1-factors is equivalent to a statement about distributions of sums of self-adjoint operators with matrix coefficients. This is an application of a linearization result for finite von Neumann…

We study selflessness in the general setting of reduced free products of $C^*$-algebras. Towards this end, we develop a suitable theory of rapid decay for filtrations in arbitrary $C^*$-probability spaces. We provide several natural…

In this paper, we study bimodules over a von Neumann algebra $M$ in two related contexts. The first is an inclusion $M \subseteq M \rtimes_\alpha G$, where $G$ is a discrete group acting on a factor $M$ by outer automorphisms. The second is…

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…

We show that Tsirelson's problem concerning the set of quantum correlations and Connes' embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchberg's QWEP conjecture) are essentially equivalent.…

We prove that every separable tracial von Neumann algebra embeds into a II$_1$ factor with property (T) which can be taken to have trivial outer automorphism and fundamental groups. We also establish an analogous result for the trivial…

We study the class of selfless C*-probability spaces introduced by Robert. It is known that a selfless tracial algebra has strict comparison and a unique trace. We prove that for separable tracial C*-algebras, selflessness is equivalent to…

The notion of a $*$-law or $*$-distribution in free probability is also known as the quantifier-free type in Farah, Hart, and Sherman's model theoretic framework for tracial von Neumann algebras. However, the full type can also be…

Given a noncommutative (nc) variety $\mathfrak{V}$ in the nc unit ball $\mathfrak{B}_d$, we consider the algebra $H^\infty(\mathfrak{V})$ of bounded nc holomorphic functions on $\mathfrak{V}$. We investigate the problem of when two algebras…