Characterized subgroups of topological abelian groups

2015/09/03 by Dikranjan, Dikran, Bruno, Anna Giordano, Impieri, Daniele
#20K45 #22A10 #22B05 #43A40 #54H11 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1509.01108

Abstract

A subgroup H of a topological abelian group X is said to be characterized by a sequence \mathbf v =(v_n) of characters of X if H=\x∈ X:v_n(x)→ 0 in \mathbb 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.

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