On the existence of topologies compatible with a group duality with predetermined properties

TL;DR

This paper investigates the conditions under which certain topologies can be imposed on abelian groups to be compatible with their duality structures, linking these conditions to properties like semireflexivity and completeness of associated topologies.

Contribution

It establishes equivalences between the existence of compatible topologies and properties like semireflexivity and completeness in the context of group dualities.

Findings

Existence of g-barrelled compatible topologies is equivalent to semireflexivity of the dual group.

Existence of k-group topologies depends on the completeness of the group's Bohr topology.

Provides criteria for constructing compatible topologies with predetermined properties.

Abstract

The paper deals with group dualities. A group duality is simply a pair $(G, H)$ where $G$ is an abstract abelian group and $H$ a subgroup of characters defined on $G$. A group topology $\tau$ defined on $G$ is {\it compatible} with the group duality (also called dual pair) $(G, H)$ if $G$ equipped with $\tau$ has dual group $H$. A topological group $(G, \tau)$ gives rise to the natural duality $(G, G^\wedge)$, where $G^\wedge$ stands for the group of continuous characters on $G$. We prove that the existence of a $g$-barrelled topology on $G$ compatible with the dual pair $(G, G^\wedge)$ is equivalent to the semireflexivity in Pontryagin's sense of the group $G^\wedge$ endowed with the pointwise convergence topology $\sigma(G^\wedge, G)$. We also deal with $k$-group topologies. We prove that the existence of $k$-group topologies on $G$ compatible with the duality $(G, G^\wedge)$ is determined by a sort of completeness property of its Bohr topology $\sigma (G, G^\wedge)$ (Theorem 3.3).