Pontryagin duality for Abelian $s$- and $sb$-groups

TL;DR

This paper explores the Pontryagin duality for Abelian s- and sb-groups, revealing conditions for dual groups, reflexivity, and characterizations of s-groups through convergent sequences.

Contribution

It establishes new duality results for Abelian s- and sb-groups, including characterizations of their duals and conditions for reflexivity and Polish properties.

Findings

A dense subgroup H of the dual group is g-closed iff it is the dual of a maximally almost periodic s-topology.

Every reflexive Polish Abelian group is g-closed in its Bohr compactification.

An s-topology generated by countable convergent sequences yields a Polish dual group.

Abstract

The main goal of the article is to study the Pontryagin duality for Abelian $s$- and $sb$-groups. Let $G$ be an infinite Abelian group and $X$ be the dual group of the discrete group $G_d$. We show that a dense subgroup $H$ of $X$ is $\mathfrak{g}$-closed iff $H$ algebraically is the dual group of $G$ endowed with some maximally almost periodic $s$-topology. Every reflexive Polish Abelian group is $\mathfrak{g}$-closed in its Bohr compactification. If a $s$-topology $\tau$ on a countably infinite Abelian group $G$ is generated by a countable set of convergent sequences, then the dual group of $(G,\tau)$ is Polish. A non-trivial Hausdorff Abelian topological group is a $s$-group iff it is a quotient group of the $s$-sum of a family of copies of $(\mathbb{Z}^\mathbb{N}_0, \mathbf{e})$.