When a totally bounded group topology is the Bohr Topology of a LCA group

TL;DR

This paper investigates when totally bounded abelian groups can be realized as the Bohr topology of locally compact abelian groups, providing necessary and sufficient conditions based on inner properties, and explores related compactness preservation issues.

Contribution

It establishes criteria for totally bounded groups to be Bohr reflections of LCA groups and provides a counterexample related to compactness preservation in MAP groups.

Findings

Criteria for totally bounded groups to be Bohr reflections of LCA groups

Counterexample of a MAP group with specific compactness properties

Answers to open questions in the literature about compactness preservation

Abstract

We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a totally bounded abelian group $(G,w)$ is the Bohr reflection of a locally compact abelian group. Necessary and sufficient conditions are established in terms of the inner properties of $w$. As an application, an example of a MAP group $(G,t)$ is given such that every closed, metrizable subgroup $N$ of $bG$ with $N \cap G = \{0\}$ preserves compactness but $(G,t)$ does not strongly respects compactness. Thereby, we respond to Questions 4.1 and 4.3 in [comftrigwu].