Maximally almost periodic groups and respecting properties

TL;DR

This paper investigates how maximally almost periodic groups respect various topological properties, characterizing the weakest property as the Schur property and exploring implications for locally convex spaces and reflexive groups.

Contribution

It introduces the concept of respecting properties in MAP groups, characterizes the Schur property as the weakest, and connects these properties to dual group structures and reflexivity.

Findings

Respecting the Schur property is the weakest among properties in rf6P.

Every real locally convex space is a quotient of one with the Schur property.

Reflexive groups respect all properties in rf6P and are Mackey groups.

Abstract

For a Tychonoff space $X$, denote by $\mathfrak{P}$ the family of topological properties $\mathcal{P}$ of being a convergent sequence or being a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of $X$, respectively. A maximally almost periodic $(MAP$) group $G$ respects $\mathcal{P}$ if $\mathcal{P}(G)=\mathcal{P}(G^+)$, where $G^+$ is the group $G$ endowed with the Bohr topology. We study relations between different respecting properties from $\mathfrak{P}$ and show that the respecting convergent sequences (=the Schur property) is the weakest one among the properties of $\mathfrak{P}$. We characterize respecting properties from $\mathfrak{P}$ in wide classes of $MAP$ topological groups including the class of metrizable $MAP$ abelian groups. Every real locally convex space (lcs) is a quotient space of an lcs with the Schur property, and every locally quasi-convex (lqc) abelian group is a quotient group of an lqc abelian group with the Schur property. It is shown that a reflexive group $G$ has the Schur property or respects compactness iff its dual group $G^\wedge$ is $c_0$-barrelled or $g$-barrelled, respectively. We prove that a locally quasi-convex abelian $k_\omega$-group respects all properties $\mathcal{P}\in\mathfrak{P}$. As an application of the obtained results we show that (1) the space $C_k(X)$ is a reflexive group for every separable metrizable space $X$, and (2) a reflexive abelian group of finite exponent is a Mackey group.