Related papers: An uncountable ergodic Roth theorem and applicatio…

Let $\mathcal{G}$ be a countably infinite group of unitary operators on a complex separable Hilbert space $H$. Let $X = \{x_{1},...,x_{r}\}$ and $Y = \{y_{1},...,y_{s}\}$ be finite subsets of $H$, $r < s$, $V_{0} = \bar{span}…

Suppose that G is a compact Abelian topological group, m is the Haar measure on G and f is a measurable function. Given (n_k), a strictly monotone increasing sequence of integers we consider the nonconventional ergodic/Birkhoff averages…

In this paper, among other things, we state and prove the mean ergodic theorem for amenable semigroup algebras.

Consider two free measure preserving group actions $\Gamma \actson (X, \mu), \Delta \actson (X, \mu)$, and a measure preserving action $\Delta \actson^a (Z, \nu)$ where $(X, \mu), (Z, \nu)$ are standard probability spaces. We show how to…

We prove a ratio ergodic theorem for amenable equivalence relations satisfying a strong form of the Besicovich covering property. We then use this result to study general non-singular actions of non-abelian free groups and establish a ratio…

Equationally compact subgroups of countable groups were introduced by Banaschewski. For all known cases the orbit closure of such a subgroup is a countable subset in the space of subgroups and has finite Cantor-Bendixson rank. We show that…

Benyamini and Lindenstrauss mention in their monograph \emph{Geometric nonlinear functional analysis Vol. 1., American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000} that there is no…

We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable…

Recently, Glasner, Lin and Meyerovitch gave a first example of a partial invariant order on a certain group that cannot be invariantly extended to an invariant random total order. Using their result as a starting point we prove that any…

We provide a pure algebraic version of the dynamical characterization of Conrad's property. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof…

We prove that, if a discrete group $G$ is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of $G$ is non-amenable with respect to the topology generated by its rank metric. This…

We use techniques of proof mining to obtain a computable and uniform rate of metastability (in the sense of Tao) for the mean ergodic theorem for a finite number of commuting linear contractive operators on a uniformly convex Banach space.

We extend previous results on noncommutative recurrence in unital *-algebras over the integers, to the case where one works over locally compact Hausdorff groups. We derive a generalization of Khintchine's recurrence theorem, as well as a…

In the local, characteristic 0, non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies that an…

We provide a framework for studying randomly coloured point sets in a locally compact, second-countable space on which a metrisable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical…

We investigate various relaxations of additivity for set maps into Banach spaces in the context of representations of amenable groups. Specifically, we establish conditions under which asymptotically additive and almost additive set maps…

For a countable abelian group $G$ we investigate generic properties of the space of all invariant metrics on $G$. We prove that for every such an unbounded group $G$, i.e. group which has elements of arbitrarily high order, there is a dense…

We generalize Luck's Theorem to show that the L^2-Betti numbers of a residually amenable covering space are the limit of the L^2-Betti numbers of a sequence of amenable covering spaces. We show that any residually amenable covering space of…

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits…

We establish a connection between two variants of van der Corput's Difference Theorem (vdCDT) for countably infinite amenable groups $G$ and the ergodic hierarchy of mixing properties of a unitary representation $U$ of $G$. In particular,…