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

Abstract

[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 τ defined on G is compatible with the group duality (also called dual pair) (G, H) if G equipped with τ has dual group H. A topological group (G, τ) gives rise to the natural duality (G, G∧), where G∧ 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∧) is equivalent to the semireflexivity in Pontryagin’s sense of the group G∧ endowed with the pointwise convergence topology σ(G∧, 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∧) is determined by a sort of completeness property of its Bohr topology σ(G, G∧) (Theorem 3.3).