Metrizable TAP, HTAP and STAP groups
The authors analyze three classes of topological groups (TAP, HTAP and STAP) defined through the nonexistence of sequences with strong convergence properties that are related to the class NSS of groups with no small subgroups. The classes HTAP and STAP are introduced in the present paper as natural strengthenings of the TAP (after \textit{trivially absolutely productive}) property introduced by \textit{D. Shakhmatov} and \textit{J. Spěvák} [Topology Appl. 157, No. 8, 1518--1540 (2010; Zbl 1195.54040)].NEWLINENEWLINEA topological group is said to be a TAP group if for every faithfully indexed sequence \((g_n)_{n\in \mathbb{N}}\) of elements of \(G\) there is a sequence \((m_k)_{n\in \mathbb{N}}\) of integers and a bijective map \(\sigma: \mathbb N\to\mathbb N\) such that the sequence \(\left(\prod_{k=1}^ng_{\sigma(k)}^{m_k}\right)_{n\in \mathbb{N}}\) is not convergent, i.e., in the terminology of the present paper, if no faithfully indexed sequence is unconditionally hyper-multipliable. For a topological group to be in HTAP it is necessary that for every faithfully indexed sequence \((g_n)_{n\in \mathbb{N}}\) of elements of \(G\) there is a sequence \((m_k)_{n\in \mathbb{N}}\) of integers such that the sequence \(\left(\prod_{k=1}^n g_{k}^{m_k}\right)_{n\in \mathbb{N}}\) is not convergent. For a topological group to be in STAP it is required that for every faithfully indexed sequence \((g_n)_{n\in \mathbb{N}}\) of elements of \(G\) there is a sequence \((m_k)_{n\in \mathbb{N}}\) of integers such that the sequence \(\left(g_n^{m_n}\right)_{n\in \mathbb{N}}\) is not convergent. It is clear from the definitions that \(\mathrm{STAP}\subset \mathrm{HTAP}\subset \mathrm{TAP}\).NEWLINENEWLINEShakhmatov and Spěvák [loc. cit.] proved that \(\mathrm{NSS}\subset \mathrm{TAP}\). In the present paper it is proved that NSS groups even have the stronger STAP property (hence \(\mathrm{NSS}\subset \mathrm{STAP}\subset \mathrm{HTAP}\subset \mathrm{TAP}\)) and that for metrizable groups the NSS and STAP properties are actually equivalent. If the group is, in addition to metrizable, Weil complete (i.e. complete in the one-sided uniformity) all four properties (NSS, STAP, HTAP and TAP) are equivalent.NEWLINENEWLINEIt is next shown that Shakhmatov and Spěvák's characterization of pseudocompactness in terms of the TAP property of \(C_p(X,\mathbb R)\), a Tychonoff space \(X\) is pseudocompact if and only if \(C_p(X,\mathbb{T})\) has the TAP property, cannot be pushed forward to the STAP property, as \(C_p(X,\mathbb R)\) has the STAP property if and only if \(X\) is finite.NEWLINENEWLINEIn the last section the authors, inspired by the results of \textit{C. Bessaga, A. Pelczynski} and \textit{S. Rolewicz} [Colloq. Math. 7, 45--51(1959; Zbl 0090.32702)] for topological vector spaces, find an adequate class of metrizable abelian groups within which a topological group has the STAP property (or, equivalently, is an NSS group) if and only if it does not contain any subgroup topologically isomorphic to \(\mathbb Z^{(\mathbb N)}\), the direct sum of countably many copies of \(\mathbb Z\). The direct sum can be replaced by the direct product \(\mathbb Z^\mathbb N\) if the group is in addition assumed to be complete. This class is denoted as MMP and contains any divisible torsion-free topological group for which the set of maps \(\{x\mapsto \frac{1}{n}x: n\in \mathbb N\}\) is equicontinuous.
scientific article; zbMATH DE number 6047960
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