A characterization of the maximally almost periodic abelian groups
For an abelian topological group \(G\), let \(\hat G\) denote its Pontryagin dual group (i.e., the group of all continuous homomorphisms of \(G\) into the circle group, \(\mathbb T\), with pointwise multiplication as composition law). Given a sequence \(\underline{u}=(u_n)\in \hat G^{\mathbb N}\), consider the subgroup \(s_{\underline{u}}(G)=\{x\in G:u_n(x)\rightarrow 0\;\text{in}\;G\}\). If \(H\) is a subgroup of \(G\), the authors define its Galois-closure (\(\mathfrak g\)-closure) by \[ \mathfrak g_G(H)=\bigcap \{s_{\underline{u}}(G):\underline{u}\in \hat G^{\mathbb N}, H\leq s_{\underline{u}}(G) \}. \] Thus, a subgroup \(H\) is \(\mathfrak g\)-{closed} if \(\mathfrak g_G(H)=H\) and \(\mathfrak g\)-\textit{dense} if \(\mathfrak g_G(H)=G\). Finally, it is said that a topological abelian group \(G\) has the \(\mathfrak g\)-{closure property} if every cyclic subgroup of \(G\) is \(\mathfrak g\)-closed. The main result of this paper is the following. \textbf{Theorem:} A topological abelian group is MAP if and only if \(G\) has the \(\mathfrak g\)-closure property. This extends a previous result which deals with the circle group and, thereby, affirms an appropriate version of a conjecture given by \textit{A. Bíró}, \textit{J.-M. Deshouillers} and \textit{V. T. Sós} in [Studia Sci. Math. Hungar. 38, 97--113 (2001; Zbl 1006.11038)]. The main result is established in several steps. In Section 2 the authors point out the basic properties of the Galois closure \(\mathfrak g_G(-)\) and reduce the problem to precompact groups. In Section 3 they analyze the structure of the subgroups \(s_{c^*}(\mathbb Z_p)\), when \(c^*\) is a subsequence of the canonical set of generators of the Prüfer group \(Z(p^\infty)\) and \(\mathbb Z_p\) denotes the group of \(p\)-adic integers. The following result of independent interest is accomplished. \textbf{Theorem:} Let \((n_k)\) be a strictly increasing sequence of naturals and let \(\underline{c^*}=(c_{n_k})\). Then the following are equivalent: (i) \(s_{c^*}(\mathbb Z_p)=\mathbb Z\); (ii) the differences \(n_{k+1}-n_k\) are bounded; (iii) the finest precompact topology \(\sigma_{\underline{c^*}}\) on \(\mathbb Z(p^\infty)\) that makes \((c_{n_k})\) converge to \(0\) is metrizable; and (iv) the finest precompact topology \(\sigma_{\underline{c^*}}\) on \(\mathbb Z(p^\infty)\) that makes \((c_{n_k})\) converge to \(0\) has weight \(<\mathfrak c\). In Section 4 it is proved that every compact abelian group has the \(\mathfrak g\)-closure property from which the main result follows. Finally, in the last Section, several interesting open questions are stated.
scientific article; zbMATH DE number 2157518
scientific article