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Characterized Subgroups of Topological Abelian Groups

Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Via delle Scienze 206, Udine 33100, Italy
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Academic Editor: Sidney A. Morris
Axioms 2015, 4(4), 459-491; https://doi.org/10.3390/axioms4040459
Received: 2 September 2015 / Revised: 27 September 2015 / Accepted: 8 October 2015 / Published: 16 October 2015
(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)
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 = {xX : 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 auto-characterized 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 proven 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. View Full-Text
Keywords: characterized subgroup; T-characterized subgroup; T-sequence; TB-sequence; von Neumann radical; convergent sequence; null sequence; abelian group characterized subgroup; T-characterized subgroup; T-sequence; TB-sequence; von Neumann radical; convergent sequence; null sequence; abelian group
MDPI and ACS Style

Dikranjan, D.; Giordano Bruno, A.; Impieri, D. Characterized Subgroups of Topological Abelian Groups. Axioms 2015, 4, 459-491. https://doi.org/10.3390/axioms4040459

AMA Style

Dikranjan D, Giordano Bruno A, Impieri D. Characterized Subgroups of Topological Abelian Groups. Axioms. 2015; 4(4):459-491. https://doi.org/10.3390/axioms4040459

Chicago/Turabian Style

Dikranjan, Dikran, Anna Giordano Bruno, and Daniele Impieri. 2015. "Characterized Subgroups of Topological Abelian Groups" Axioms 4, no. 4: 459-491. https://doi.org/10.3390/axioms4040459

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