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Duality for Convergence Abelian Groups
by
M. Montserrat Bruguera & M. Jesús Chasco & Elena Martín-Peinador
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A number of attempts to extend Pontryagin duality theory to categories of groups larger than that of locally compact abelian groups have been made using different approaches. The extension to the category of topological abelian groups created the concept of reflexive group. We deal now with the extension of Pontryagin duality to the category of convergence abelian groups. Reflexivity in this category was defined and studied by E. Binz and H. Butzmann. A convergence group is reflexive (subsequently called BB-reflexive by us in our work) if the canonical embedding into the bidual is a convergence isomorphism.
We denote by \Gamma G the set of all continuous homomorphisms (i.e. continuous characters) from an abelian topological group G into the circle group \Bbb T . If addition in \Gamma G is defined pointwise, then \Gamma G endowed with the compact open topology is a topological abelian group, which will be called G\wedge .
The group G is called reflexive if the natural embedding \alphaG from G into the bidual G\wedge\wedge:=(G\wedge)\wedge is a topological isomorphism. The classical Pontryagin duality theorem states that every locally compact topological abelian group (LCA) is reflexive.
Examples of reflexive groups which are not locally compact are known from the late forties. In [7] it is proved that arbitrary products of LCA-groups are reflexive, whilst they may not be locally compact, like {\Bbb R}\omega or {\Bbb R}c . In [9] it is proved that any infinite dimensional Banach space considered in its additive structure is a reflexive group.
We include here the definition of a convergence structure, and of a convergence group.
Let X be a set and suppose that to each x in X is associated a collection \Xi (x) of filters on X satisfying:
The totality \Xi of filters \Xi(x) for x in X is called a convergence structure for X , the pair (X,\Xi) a convergence space and the filters {\Cal F} in \Xi(x) will be called convergent to x . We write {\Cal F}--> x instead of {\Cal F} in \Xi(x) . A mapping f:X--> Y between two convergence spaces X and Y is continuous if f({\Cal F})--> f(x) in Y whenever {\Cal F}--> x in X .
A convergence group (G,\Xi) , or briefly G , is a group for which the convergence structure \Xi is compatible with addition. If G is a convergence group, we also call \Gamma G the set of all continuous homomorphisms (in the sense of convergence) from G into \Bbb T and the continuous convergence structure \Lambda , in \Gamma G , is defined in the following way: a filter \Cal F in \Gamma G converges in \Lambda to an element \xi in \Gamma G if for every x in G and every filter \Cal H in G , convergent to x , w({\Cal F}× {\Cal H}) converges to \xi(x) in \Bbb T . ({\Cal F}× {\Cal H} denotes the filter generated by the products F× G , F\in{\Cal F} and H\in{\Cal H} , and w(F× H):= {f(x); f in F and x in H} ). Thus \Lambda is the coarsest convergence structure in \Gamma G for which the evaluation mapping w is continuous.
E. Binz and H. Butzmann have succeeded to extend Pontryagin duality theory to the category of convergence abelian groups and continuous homomorphisms, CONABGRP. They first define the "convergence dual" \Gamma c G of a group G \in CONABGRP, as the set of all continuous characters endowed with the "continuous convergence structure". If G is a LCA group, the continuous convergence structure in \Gamma G is precisely the convergence given by the compact open topology [3], thus, the "convergence dual" and the ordinary dual are identical. They call G reflexive if the natural embedding \kappaG : G --> \Gammac \Gammac G is an isomorphism in the category CONABGRP. They have studied many features of this concept of reflexivity. To mention one, a topological vector space, regarded as an abelian group, is BB-reflexive if and only if it is locally convex and complete [4].
Topological abelian groups are, in an obvious way, convergence groups, therefore it is natural to compare reflexivity and BB-reflexivity for them. We have proved that these two notions are in general independent although they coincide for some classes of topological groups, for instance for metrizable groups.
A natural question is to study properties of reflexive groups shared also by BB-reflexive groups. In previous work [2] we proved the following:
1) If A is an open subgroup of a topological group G , then G is reflexive if and only if A is reflexive.
2) If K is a compact subgroup of a group G with sufficiently many continuous characters, then G is reflexive iff G/K is reflexive. We have seen that analogous statements hold for BB-reflexivity.
Finally we have used the continuous convergence structure to prove that every reflexive admissible topological group must be locally compact.
\Refs
\ref\key 1 \by Banaszczyk, W. \paper On the existence of Exotic Banach-Lie Groups \jour Math. Ann. \vol 264 \yr 1983 \pages 485-493\endref
\ref\key 2 \by Banaszczyk, W. - Chasco, M.J. - Mart\'{\i}n-Peinador, E. \paper Open Subgroups and Pontryagin Duality \jour Mathematische Zeitschrift \vol 215 \yr 1994 \pages 195-204\endref
\ref\key 3 \by Binz, E. \paper Continuous Convergence in C(X) \jour Lecture Notes in Mathematics 469. Springer-\-Ver\-lag, Berlin Heidelberg New York, 1975 \endref
\ref\key 4 \by Butzmann, H.P. \paper Pontrjagin-Dualität für topologische Vektorräume \jour Arch. Math. \vol 28 \yr 1977 \pages 632-637\endref
\ref\key 5 \by Chasco, M.J. - Mart\'{\i}n-Peinador, E. \paper Binz-Butzmann duality versus Pontryagin Duality \jour Archiv der Math. \vol 63 \issue No.3 \yr 1994 \pages 264-270\endref
\ref\key 6 \by Fischer, H.R. \paper Limesräume \jour Math. Ann. \vol 137 \yr 1959 \pages 269-303\endref
\ref\key 7 \by Kaplan, S. \paper Extension of the Pontryagin Duality. I: Infinite Products \jour Duke Math. J. \vol 15 \yr 1948 \pages 649-658\endref
\ref\key 8 \by Mart\'{\i}n-Peinador, E. \paper A reflexive admissible topological group must be locally compact \jour Proc. Amer. Math. Soc. \vol 123 \yr 1995 \pages 3563-3566\endref
\ref\key 9 \by Smith Freundlich, M. \paper The Pontryagin duality theorem in linear spaces \jour Ann. of Math. \vol (2) 56 \yr 1952 \pages 248-253\endref