Related papers: Noncommutative recurrence over locally compact Hau…

We consider an action of a compact group whose dual is archimedean linearly ordered or a direct product (or sum) of such groups on a von Neumann algebra, M. We define the generalized Hardy subspace of the Hilbert space of a standard…

We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex p\{i(\lambda_k^{(1)}t_1+...+\lambda_k^{(s)}t_s)\}$, where $\{X_n\}$ is a sequence of random variables and $\{\lambda_n^{(i)}:1\leq i\leq s\}$ are sequences of real…

We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary…

We present a generalization of Hirschman's entropic uncertainty principle for locally compact abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups,…

For $A$ a $C^*$-algebra, $E_1, E_2$ two Hilbert bimodules over $A$, and a fixed isomorphism $\chi : E_1\otimes_AE_2\to E_2\otimes_AE_1$, we consider the problem of computing the $K$-theory of the Cuntz-Pimsner algebra ${\mathcal…

We generalize to the setting of Arveson's maximal subdiagonal subalgebras of finite von Neumann algebras, the Szeg\"o $L^p$-distance estimate, and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. In so doing, we…

For any topological space there is a sheaf cohomology. A Grothendieck topology is a generalization of the classical topology such that it also possesses a sheaf cohomology. On the other hand any noncommutative $C^*$-algebra is a…

We study criteria for a ring - or more generally, for a small category - to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to…

We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain…

Given an ample Hausdorff groupoid $G$, a unital commutative ring $R$, and a discrete twist $(\Sigma,i,q)$, we establish a generalised uniqueness theorem for the twisted Steinberg algebra $A_R(G;\Sigma)$. By applying this theorem when $G$ is…

The aim of the article is to provide characterizations of the Haage-rup property for locally compact, second countable groups in terms of approximations of some non-ergodic invariant states by mixing ones for actions on unital…

A weight-dependent generalization of the binomial theorem for noncommuting variables is presented. This result extends the well-known binomial theorem for q-commuting variables by a generic weight function depending on two integers. For a…

We introduce a definition of the locally trivial $G$-C*-algebra, which is a noncommutative counterpart of the total space of a locally compact Hausdorff numerable principal $G$-bundle. To obtain this generalization, we have to go beyond the…

In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain first and second order differential equations for the orthogonal polynomials and associated functions with a weight on the unit circle. We…

The principal results proved in this expository thesis are the IP polynomial Szemer\'edi theorem for nilpotent groups and the multiple term return times theorem with nilsequence weights. It also contains extensions of the convergence…

Let $U$ be an algebraic subgroup of the group of $n\times n$ upper-triangular matrices with units on the diagonal over a finite field of large enough characteristic, and $\mathfrak{n}$ be the Lie algebra of $U$. The main tool in…

We consider a version of the notion of F-inverse semigroup (studied in the algebraic theory of inverse semigroups). We point out that an action of such an inverse semigroup on a locally compact space has associated a natural groupoid…

We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements…

We construct the full and reduced C*-algebras of an ample groupoid from its complex Steinberg algebra. We also show that our construction gives the same C*-algebras as the standard constructions. In the last section, we consider an…

The space of deformations of the integer Heisenberg group under the action of $\textrm{Aut}(H(\mathbb{R}))$ is a homogeneous space for a non-reductive group. We analyze its structure as a measurable dynamical system and obtain mean and…