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On strongly reflexive topological groups

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2001
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Universidad Politécnica de Valencia
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Let Gˆ denote the Pontryagin dual of an abelian topological group G. Then G is reflexive if it is topologically isomorphic to Gˆˆ, strongly reflexive if every closed subgroup and every Hausdorff quotient of G and of Gˆ is reflexive. It is well known that locally compact abelian (LCA) groups are strongly reflexive. W. Banaszczyk [Colloq. Math. 59 (1990), no. 1, 53–57], extending an earlier result of R. Brown, P. J. Higgins and S. A. Morris [Math. Proc. Cambridge Philos. Soc. 78 (1975), 19–32], showed that all countable products and sums of LCA groups are strongly reflexive. L. Aussenhofer [Dissertationes Math. (Rozprawy Mat.) 384 (1999), 113 pp.] showed that all Čech-complete nuclear groups are strongly reflexive. It is an open question whether the strongly reflexive groups are exactly the Čech-complete nuclear groups and their duals. A Hausdorff topological group G is almost metrizable if and only if it has a compact subgroup K such that G/K is metrizable [W. Roelcke and S. Dierolf, Uniform structures on topological groups and their quotients, McGraw-Hill, New York, 1981]. In this paper it is shown that the annihilator of a closed subgroup of an almost metrizable group G is topologically isomorphic to the dual of the corresponding Hausdorff quotient, and an analogous statement holds for the character group of G. It then follows that an almost metrizable group is strongly reflexive only if its Hausdorff quotients and those of its dual are reflexive.
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L. Aussenhofer, Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Dissertationes Math. (Rozprawy Mat.) 384 (1999), 113 Maria Banaszczyk and Wojciech Banaszczyk, Characterization of nuclear spaces by means of additive subgroups, Math. Z. 186 (1984), no. 1, 125-133 W. Banszczyk, Countable products of LCA groups: their closed subgroups, quotients and duality properties, Colloq. Math. 59 (1990), no. 1, 53-57 Wojciech Banaszczyk, Additive subgroups of topological vector spaces, Springer-Verlag, Berlin, 1991 Ronald Brown, Philip J. Higgins, and Sidney A. Morris, Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties, Math. Proc. Cambridge Philos. Soc. 78 (1975), 19-32 M. Bruguera, Grupos topológicos y grupos de convergencia: estudio de la dualidad de Pontryagin, Doctoral dissertation, Barcelona, 1999 M. J. Chasco, Pontryagin duality for metrizable groups, Arch. Math. (Basel) 70 (1998), no. 1, 22-28 Samuel Kaplan, Extensions of the Pontrjagin duality. I. Infinite products, Duke Math. J. 15 (1948), 649-658 J. Margalef, E. Outerelo, and J. L. Pinilla, Topología V, Ed. Alhambra, 1979 N. Noble, k-groups and duality, Trans. Amer. Math. Soc. 151 (1970), 551-561 Walter Roelcke and Susanne Dierolf, Uniform structures on topological groups and their quotients, McGraw-Hill International Book Co., New York, 1981 N. Ya. Vilenkin, The theory of characters of topological Abelian groups with boundedness given, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 439-462
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