Related papers: Unique prime factorization and bicentralizer probl…

We prove that, under the continuum hypothesis $\frak c=\aleph_1$, any ultraproduct II$_1$ factor $M= \prod_{\omega} M_n$ of separable finite factors $M_n$ contains more than $\frak c$ many mutually disjoint singular MASAs, in other words…

Building on Lin's breakthrough MIP$^{co}$ = coRE and an encoding of non-local games as universal sentences in the language of tracial von Neumann algebras, we show that locally universal tracial von Neumann algebras have undecidable…

We compute the $K_1$-group of ultraproducts of unital, simple $C^*$-algebras with unique trace and strict comparison. As an application, we prove that the reduced free group $C^*$-algebras $C^*_r(F_m)$ and $C^*_r(F_n)$ are elementarily…

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 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…

In this paper we prove two results. The first theorem uses a paper of Kim \cite{K} to show that for fixed primes $p_1,...,p_k$, and for fixed integers $m_1,...,m_k$, with $p_i\not|m_i$, the numbers $(e_{p_1}(n),...,e_{p_k}(n))$ are…

We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes Embedding Problem is self-tracially stable if and only if it is amenable. We then generalize a…

We provide a class of separable II$_1$ factors $M$ whose central sequence algebra is not the "tail" algebra associated to any decreasing sequence of von Neumann subalgebras of $M$. This settles a question of McDuff \cite{Mc69d}.

Type III_1 factors arising as (direct summands of) von Neumann algebraic free products are investigated. In particular we compute Connes' Sd- and tau- invariants for those type III_1 factors without any extra assumption.

We give a short and simple proof, utilizing the pre-determinant of P. de la Harpe and G. Skandalis, that the universal covering group of the unitary group of a II$_1$ von Neumann algebra $\mathcal{M}$, when equipped with the norm topology,…

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free…

We study the von Neumann algebra, generated by the regular representations of the infinite-dimensional nilpotent group $B_0^{\mathbb Z}$. In [14] a condition have been found on the measure for the right von Neumann algebra to be the…

We obtain new Bass-Serre type rigidity results for ${\rm II_1}$ equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show…

We study correspondences of tracial von Neumann algebras from the model-theoretic point of view. We introduce and study an ultraproduct of correspondences and use this ultraproduct to prove, for a fixed pair of tracial von Neumann algebras…

Let $M$ be a finite von Neumann algebra (resp. a type II$_{1}$ factor) and let $N\subset M$ be a II$_{1}$ factor (resp. $N\subset M$ have an atomic part). We prove that the inclusion $N\subset M$ is amenable implies the identity map on $M$…

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 prove rigidity properties for von Neumann algebraic graph products. We introduce the notion of rigid graphs and define a class of II$_1$-factors named $\mathcal{C}_{\rm Rigid}$. For von Neumann algebras in this class we show a unique…

In this paper we analyze the structure of some sets of non-commutative moments of elements in a finite von Neumann algebra M. If the fundamental group of M is R_+\{0}, then the moment sets are convex, and if M is isomorphic to M tensor M,…

We prove that certain free products of factors of type ${\rm I}$ and other von Neumann algebras with respect to nontracial, almost periodic states are almost periodic free Araki-Woods factors. In particular, they have the free absorption…

We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain properties, which ensure that a II$_1$ factor $M$ is prime or has at most one Cartan subalgebra. For instance, we prove that if there…