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

• Published in: arXiv.org 2021-06
• Main Authors: Borsich, Tayomara, Domínguez, Xabier, Martín-Peinador, Elena
• Format: Article
• Language: English
• Subjects: Compatibility; Group theory; Subgroups; Topology
• ISSN: 2331-8422

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).