Arcs in the Pontryagin dual of a topological abelian group
Let \(G\) be a topological abelian group. The dual group \(\widehat G\) of \(G\) is defined as the group of all continuous homomorphisms (characters) of \(G\) into the unit circle of the complex plane \(\mathbb S^1\) and it is equipped with the compact open topology. With this topology \(\widehat G\) is a Hausdorff topological abelian group, which is generally called Pontryagin dual of \(G\). Let \(\widehat G_a\) denote the arc-component of \(\widehat G\), \(\widehat G_{\mathrm{lift}}\) the group of all characters which can be lifted over the real numbers \(\mathbb R\), and \(\widehat G_{\mathrm{lift}\sigma}\) the group of those characters that can be lifted over \(\mathbb R\) equipped with its Bohr topology (the topology of pointwise convergence on \(\widehat {\mathbb R}\)). If \(\exp_G\) denotes the exponential function of \(G\), the elements of its image \(\mathrm{im} \exp_G\) are those lying in one-parameter subgroups, and \(G\) is the union of its one-parameter subgroups if and only if \(\exp_G\) is onto. Among others, the following results are proved: (1) in Theorem 3.6, it is established that \(G_{\mathrm{lift}}\subseteq G_{\mathrm{lift}\sigma}\subseteq \mathrm{im} \exp_{\widehat G}\subseteq \widehat G_a\). Furthermore, in analyzing conditions under which these equalities hold, the authors introduce the equicontinuous arc property (EAP for short), which is satisfied for a topological abelian group \(G\) if every arc in \(\widehat G\) is equicontinuous. In Theorem 3.13, it is proved that if \(G\) has EAP, then \(G_{\mathrm{lift}}= \widehat G_a\) holds. The paper also contains other interesting results and pertinent examples.
scientific article; zbMATH DE number 6387123
scientific article