Characterization of almost maximally almost-periodic groups

TL;DR

This paper characterizes abelian groups with certain topological properties related to their von Neumann radicals, showing conditions under which these radicals are non-trivial, finite, and how they behave under various topologies.

Contribution

It provides a complete characterization of when abelian groups admit topologies with non-trivial finite von Neumann radicals and analyzes the properties of these radicals in topological groups.

Findings

Groups admit non-trivial finite von Neumann radicals iff they have non-trivial finite subgroups.

The von Neumann radical of the integers is invariant under any Hausdorff topology.

Conditions for the radical's behavior under dual embedding are established.

Abstract

Let $G$ be an abelian group. We prove that a group $G$ admits a Hausdorff group topology $\tau$ such that the von Neumann radical $\mathbf{n}(G, \tau)$ of $(G, \tau)$ is non-trivial and finite iff $G$ has a non-trivial finite subgroup. If $G$ is a topological group, then $\mathbf{n} (\mathbf{n} (G)) \not= \mathbf{n} (G)$ if and only if $\mathbf{n} (G)$ is not dually embedded. In particular, $\mathbf{n} (\mathbf{n} (\mathbb{Z},\tau)) = \mathbf{n} (\mathbb{Z},\tau)$ for any Hausdorff group topology $\tau$ on $\mathbb{Z}$.