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

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).