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A survey on strong reflexivity of abelian topological groups

Martín Peinador, Elena and Chasco, M.J. (2012) A survey on strong reflexivity of abelian topological groups. Topology and its Applications . ISSN 0166-8641 (Submitted)

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The Pontryagin duality Theorem for locally compact abelian groups (briey, LCA-groups)
has been the starting point for many different routes of research in Mathematics. From its appearance there was a big interest to obtain a similar result in a context broader than LCA-groups. Kaplan in the 40's proposed -and it remains open- the characterization of all the abelian topological groups for which the canonical mapping into its bidual is a topological isomorphism, assuming that the dual and the bidual carry the compact-open topology. Such groups are called reflexive.
In this survey we deal with results on reflexivity of certain classes of groups, with special emphasis on the class which better reflects the properties of LCA-groups, namely that of strongly reflexive groups. A topological abelian group is said to be strongly reflexive if all its closed subgroups and its Hausdorff quotients as well as the closed subgroups and the Hausdorff quotients of its dual group are reflexive.
By no means we can claim completeness of the survey: just an ordered view of the topic, with some small new results indicated in the text

Item Type:Article
Uncontrolled Keywords:Pontryagin duality theorem, Dual group, Reflexive group, Strongly reflexive group, Metrizable group, Čech-complete group, ω-bounded group, P-group
Subjects:Sciences > Mathematics > Topology
ID Code:15139

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