Characterized Subgroups of Topological Abelian Groups

Abstract

A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {x ∈ X : vn(x) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of autochacaracterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proved to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups.

Characterized subgroups of the circle group

“…For other results on characterized subgroups and related topics see the recent papers [38,31,32,33,35,36,37,52,53,54,55,70].…”

“…Both "characterized subgroups" and "topologically u-torsion subgroups" can be successfully defined and used for arbitrary topological abelian groups; we dedicate some attention to these more general versions in § §7,8, trying to show in §8 how big is the divergence between these two approaches; §8 contains also a non-abelian version of "topologically u-torsion element". The non-abelian counterpart of "characterized subgroup" of a compact group is largely discussed in the nice survey paper [54], while characterized subgroups of arbitrary topological abelian groups are discussed in the survey paper [33].…”