Entropy Range Problems and Actions of Locally Normal Groups

Measures of maximal entropy

We extend the results of Walters on the uniqueness of invariant measures with maximal entropy on compact groups to an arbitrary locally compact group. We show that the maximal entropy is attained at the left Haar measure and the measure of maximal entropy is unique.

Shift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups

We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...

C∗-algebras and topological dynamics: finite approximation and paradoxicality

Contents Chapter 1. Introduction 5 Chapter 2. Internal measure-theoretic phenomena 13 1. Amenable groups and nuclearity 13 2. Amenable actions, nuclearity, and exactness 16 3. The type semigroup, invariant measures, and pure infiniteness 21 4. The universal minimal system 29 5. Minimal actions, pure infiniteness, and nuclearity 31 Chapter 3. External measure-theoretic phenomena 33 1. Sofic grou...

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras

Diagonalizable flows on locally homogeneous spaces and number theory

We discuss dynamical properties of actions of diagonalizable groups on locally homogeneous spaces, particularly their invariant measures, and present some number theoretic and spectral applications. Entropy plays a key role in the study of theses invariant measures and in the applications. Mathematics Subject Classification (2000). 37D40, 37A45, 11J13, 81Q50.