Maximally almost periodic groups and respecting properties: In honour of manuel lópez-pellicer

A maximally almost periodic topological (MAP) group G respects p if p(G)=p(G⁺), where G⁺ is the group G endowed with the Bohr topology and p stands for the subsets of G that have the property p. For a Tychonoff space X, we denote by (formula presented) the family of topological properties p of being a convergent sequence or a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of X, respectively. We study relations between different respecting properties from (formula presented) and show that the respecting convergent sequences (=the Schur property) is the weakest one among the properties of (formula presented). We characterize respecting properties from (formula presented) 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 (formula presented) is c0 -barrelled or g-barrelled, respectively. We prove that an lqc abelian K_ω-group respects all properties (formula presented). As an application of the obtained results we show that a reflexive abelian group of finite exponent is a Mackey group.