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Pontryagin duality for metrizable groups. (English) Zbl 0899.22001
An abelian topological group \(G\) is called reflexive if the canonical mapping from \(G\) to the second dual group \(\nu\colon G^{\wedge\wedge}\) is an isomorphism of topological groups, where \(G^\wedge\) is the group of continuous characters of \(G\) equipped with the compact-open topology. The classical result by Pontryagin-van Kampen says that every locally compact abelian (LCA) group is reflexive. A lot of effort has been made over the past decades to try and find natural classes of reflexive non-LCA topological abelian groups. Perhaps the best known achievements are those by Smith who proved the reflexivity of additive groups of Banach spaces [M. Smith, Ann. Math., II. Ser. 56, 248-253 (1952; Zbl 0047.10701)], by Kaplan who proved that the class of reflexive groups is closed under some natural operations not preserving local compactness [S. Kaplan, Duke Math. J. 15, 649-658 (1948; Zbl 0034.30601)], and by Banaszczyk, who had introduced the class of so-called nuclear topological groups and shown that every complete metrizable nuclear group is reflexive [W. Banaszczyk, Additive subgroups of topological vector spaces, Lect. Notes Math. 1466 (Berlin etc. 1991; Zbl 0743.46002)]. The present reviewer had proved in [V. Pestov, Bull. Aust. Math. Soc. 52, 297-311 (1995; Zbl 0841.22001)] that free topological abelian groups over some topological spaces are reflexive, and asked whether every complete and Čech-complete topological group having sufficiently many characters is reflexive. The present article provides, among many other things, a negative answer to this question: it is shown that every complete locally bounded non locally convex topological vector space admitting a weaker locally convex topology (such as \(l_p\) with \(0<p<1\)) supplies a counter-example. The central result of the note under review is however this: a metrizable separable (that is, Polish) abelian group is reflexive if and only if \(\nu\colon G^{\wedge\wedge}\) is an algebraic isomorphism. It is also shown that a metrizable topological abelian group is reflexive if and only if it satisfies the duality for convergence groups introduced and studied by Binz and Butzmann [E. Binz, Continuous convergence on \(C(X)\), Lect. Notes Math. 469 (Berlin etc. 1975; Zbl 0306.54003)], even if in general the two dualities are known to be different [M. J. Chasco and E. Martin-Peinador, Arch. Math. 63, 264-270 (1994; Zbl 0806.43003)].

MSC:
22A05 Structure of general topological groups
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
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