On the existence of topologies compatible with a group duality 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).