On strongly reflexive topological groups

Abstract

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.