Publication: Open subgroups and Pontryagin duality
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Publication Date
1994-01
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Springer
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For an abelian topological group G, let G∧ denote the character group of G. The group G is called reflexive if the evaluation map is a topological isomorphism of G onto G∧∧, and G is called strongly reflexive if all closed subgroups and quotient groups of G and G∧ are reflexive. In this paper the authors study the relationship of reflexivity (and strong reflexivity) among G, A, and G/K, where A is an open subgroup and K a compact subgroup of G. Strong reflexivity is closely connected with the notion of strong duality introduced by R. Brown, P. J. Higgins and S. A. Morris [Math. Proc. Cambridge Philos. Soc. 78 (1975), 19–32]. In fact, G is strongly reflexive if and only if the natural homomorphism G∧×G→T is a strong duality. R. Venkataraman [Math. Z. 143 (1975), no. 2, 105–112] originally claimed that if G is reflexive, then so is A. However, his proof includes inaccuracies. The present paper includes a new proof in this regard. In all, the following theorems are proved. Theorem 1: G is reflexive [resp. strongly reflexive] if and only if A is reflexive [resp. strongly reflexive]. Theorem 2: If G admits sufficiently many continuous characters and G/K is reflexive [resp. strongly reflexive], then G is reflexive [resp. strongly reflexive]. Conversely, if G is reflexive and K is dually closed in G, then G/K is reflexive. Theorem 3: Every closed subgroup H and the quotient group G/H of a strongly reflexive group G are strongly reflexive
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Banaszczyk, W.: Additive subgroups of topological vector spaces. (Lect. Notes Math., vol. 1466) Berlin Heidelberg New York: Springer 1991
Brown, R., Higgins, P.J., Morris, S.A.: Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties. Math. Proc. Camb. Philos. Soc.78, 19–32 (1975)
Kaplan, S.: Extensions of the Pontrjagin duality. I. Infinite products. Duke Math. J.15, 649–658 (1948)
Noble, N.:k-groups and duality. Trans. Am. Math. Soc.151, 551–561 (1970)
Venkataraman, R.: Extensions of Pontryagin duality. Math. Z.143, 105–112 (1975)