Related papers: Tensor product indecomposability results for exist…

We prove that there exist uncountably many separable II$_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II$_1$ factors originally introduced by…

We prove that if C is a tensor C*-category in a certain class, then there exists an uncountable family of pairwise non stably isomorphic II_1 factors (M_i) such that the bimodule category of M_i is equivalent to C for all i. In particular,…

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…

A variant of Gromov's notion of measure equivalence for groups has been introduced for II$_1$ factors under different names. We propose the terminology of W*-correlated II$_1$ factors. We prove rigidity results up to W*-correlations for…

We provide a fairly large class of II$_1$ factors $N$ such that $M=N\bar{\otimes}R$ has a unique McDuff decomposition, up to isomorphism, where $R$ denotes the hyperfinite II$_1$ factor. This class includes all II$_1$ factors…

The literature contains interesting examples of inclusions of simple C$^*$-algebras with the property that all intermediate C$^*$-algebras likewise are simple. In this article we take up a systematic study of such inclusions, which we refer…

In this paper we show various new structural properties of free group factors using the recent resolution (due independently to Belinschi-Capitaine and Bordenave-Collins) of the Peterson-Thom conjecture. These results include the resolution…

An open question of Sorin Popa asks whether or not every $R^{\mathcal{U}}$-embeddable factor admits an embedding into $R^{\mathcal{U}}$ with factorial relative commutant. We show that there is a locally universal McDuff II$_1$ factor $M$…

We prove some unique factorization results for tensor products of free quantum group factors. They are type III analogues of factorization results for direct products of bi-exact groups established by Ozawa and Popa. In the proof, we first…

Using the tensor identity, we obtain decomposition results for the tensor product of a generalized Verma module with a module $M$ in the category $\mathcal{O}^{\mathfrak{p}}$, based on the decomposition of the restriction of $M$ to the…

We characterise the existentially closed models of the theory of exponential fields. They do not form an elementary class, but can be studied using positive logic. We find the amalgamation bases and characterise the types over them. We…

In the mid thirties Murray and von Neumann found a natural way to associate a von Neumann algebra $L(\Gamma)$ to any countable discrete group $\Gamma$. Classifying $L(\Gamma)$ in term of $\Gamma$ is a notoriously complex problem as in…

We investigate $E_0-$semigroups on general factors, which are not necessarily of type I, and analyse associated invariants like product systems, super product systems etc. By tensoring $E_0-$semigroups on type I factors with…

We study a family of determinantal ideals whose decompositions encode the structural zeros in conditional independence models with hidden variables. We provide explicit decompositions of these ideals and, for certain subclasses of models,…

Motivated by examples in infinite group theory, we classify the finite groups whose subgroups can never be decomposed as direct products.

A collection of 50 open problems around the structure theory of ultraproducts of II$_1$ factors is presented, along with some annotations and references.

Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C*-algebra are defined. Applications of these tracial invariants include: (1) An analogue of Szego's limit theorem for arbitrary…

We introduce and study the definition, main properties and applications of iterated twisted tensor products of algebras, motivated by the problem of defining a suitable representative for the product of spaces in noncommutative geometry. We…

This thesis examins a generalisation of polar decompositions to indefinite inner product spaces. The necessary general theory is studied and some general results are given. The main part of the thesis focuses on polar decompositions with…

We consider II$_1$ factors $M$ which can be realized as inductive limits of subfactors, $N_n \nearrow M$, having spectral gap in $M$ and satisfying the bi-commutant condition $(N_n'\cap M)'\cap M=N_n$. Examples are the enveloping algebras…