Compatible group topologies on a locally quasi-convex abelian group and the Mackey group problem

TL;DR

This paper characterizes all compatible topologies on locally quasi-convex abelian groups and explores the Mackey group problem, providing new insights into Mackey groups and their topological structures.

Contribution

It offers the first comprehensive description of compatible group topologies on lqc abelian groups and characterizes Mackey groups and their products.

Findings

Characterization of all compatible topologies on lqc abelian groups

Identification of which lqc groups are Mackey groups

Analysis of Mackey properties in product groups

Abstract

For a locally quasi-convex (lqc) abelian group $G$, we give the first description of all compatible group topologies on $G$ and apply this result to the Mackey group problem for lqc groups. We characterize lqc abelian groups which are Mackey groups or admit a Mackey group topology and provide a characterization of two Mackey groups whose product is Mackey. We obtain the first characterization of locally convex spaces which are Mackey groups.