We prove that the group-measure-space von Neumann algebra L(T) ⋊ SL(2,Z) is solid. The proof uses topological amenability of the action of SL(2,Z) on the Higson corona of Z.

Let m be a Radon measure on a Hausdorff topological space X. Corresponding to three kinds of outer measures, three kinds of m-negligible sets are considered. The main theorem states that in a metacompact space X each locally m-negligible set is m-negligible. For a Radon measure in a HausdorfT topological space X we distinguish three kinds of negligible sets corresponding to three kinds of outer...

We consider the Fröbenius-Perron semigroup of linear operators associated to a semidynamical system defined in a topological space X endowed with a finite or a σ -finite regular measure. We prove that if there exists a faithful invariant measure for the semidynamical system, then the Fröbenius-Perron semigroup of linear operators is C0-continuous in the space Lμ(X). We also give a geometrical c...

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational princi...

This survey is focused on the results related to topologies on the groups of transformations in ergodic theory, Borel, and Cantor dynamics. Various topological properties (density, connectedness, genericity) of these groups and their subgroups are studied. In this paper, we intend to present a unified approach to the study of topological properties of transformation groups in ergodic theory, Bo...

We study topological von Neumann regularity and principal von Neumann regularity of Banach algebras. Our main objective is comparing these two types of Banach algebras and some other known Banach algebras with one another. In particular, we show that the class of topologically von Neumann regular Banach algebras contains all $C^*$-algebras, group algebras of compact abelian groups and ...

In this paper, we first present a new type of the concept of open sets by expressing some properties of arbitrary mappings on a power set. With the generalization of the closure spaces in categorical topology, we introduce the generalized topological spaces and the concept of generalized continuity and become familiar with weak and strong structures for generalized topological spaces. Then, int...

We show the space of expanding Blaschke products on S is compactified by a sphere of invariant measures, reminiscent of the sphere of geodesic currents for a hyperbolic surface. More generally, we develop a dynamical compactification for the Teichmüller space of all measure-preserving topological covering maps of S.

In this paper we introduce and study so-called k∗-metrizable spaces forming a new class of generalized metric spaces, and display various applications of such spaces in topological algebra, functional analysis, and measure theory. By definition, a Hausdorff topological space X is k∗-metrizable if X is the image of a metrizable space M under a continuous map f : M → X having a section s : X → M ...