Maximally almost periodic groups and respecting properties

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.