Straightening theorem for bounded Abelian groups
For any abelian group \(G\), let \(G^{\sharp}\) denote the group \(G\) equipped with its Bohr topology, i. e. the initial topology of \(G\) with respect to the family of all homomorphisms of \(G\) into the circle group. For a positive integer \(m\) and a cardinal \(\kappa,\) let \({\mathbb V}_m^{\kappa}=\bigoplus_{\kappa} {\mathbb Z_m},\) where \({\mathbb Z}_m\) denotes the cyclic group of order \(m\). The authors prove that for any bounded abelian group \(J\), any \(m>1\) and any continuous function \(\pi:({\mathbb V_m^{\mathbb N}})^{\sharp}\to J^{\sharp},\) there exists an infinite subset \(A\) of \({\mathbb N}\) and a homomorphism \(\ell: {\mathbb V_m^{\mathbb N}}\to J\) such that \(\pi\) coincides with \(\ell\) on the set of all elements of \({\mathbb V_m^A}\) with at most \(m\) nonzero coordinates. Using the Prüfer structure theorem, this result may be briefly put as follows: every continuous function between two bounded abelian groups endowed with their Bohr topologies coincides with a homomorphism when restricted to an infinite subset of the domain. This extends results by \textit{K. Kunen} [Topology Appl. 90, 97--107 (1998, Zbl 0974.54025)] and by \textit{D. Dikranjan} and \textit{S. Watson} [J. Pure Appl. Algebra 163, 147--158 (2001; Zbl 0983.22001)]. A number of applications of this ``straightening'' theorem, mainly concerning Bohr homeomorphisms, is given.
scientific article; zbMATH DE number 5365231