A Halmos-von Neumann theorem for actions of general groups

TL;DR

This paper introduces a new categorical framework for the Halmos-von Neumann theorem, establishing equivalences between different types of dynamical systems with discrete spectrum and deriving classification results using duality theories.

Contribution

It provides a novel categorical approach to the theorem for general groups and connects topological and measure-preserving systems through duality, offering new classification tools.

Findings

Categories of topological and measure-preserving systems with discrete spectrum are equivalent.

The framework yields a complete isomorphism invariant for compactifications of topological groups.

Classification results are obtained via Pontryagin and Tannaka-Krein duality theories.

Abstract

We give a new categorical approach to the Halmos-von Neumann theorem for actions of general topological groups. As a first step, we establish that the categories of topological and measure-preserving irreducible systems with discrete spectrum are equivalent. This allows to prove the Halmos-von Neumann theorem in the framework of topological dynamics. We then use the Pontryagin and Tannaka-Krein duality theories to obtain classification results for topological and then measure-preserving systems with discrete spectrum. As a byproduct, we obtain a complete isomorphism invariant for compactifications of a fixed topological group.