Duality for Convergence Abelian Groups
by
M. Montserrat Bruguera
Universidad Politécnica de Cataluña
Coauthors: M. Jesús Chasco (Universidad de Vigo), Elena Martín-Peinador (Universidad Complutense de Madrid)

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 \GammaG the set of all continuous homomorphisms (i.e. continuous characters) from an abelian topological group G into the circle group T. If addition in \GammaG is defined pointwise, then \GammaG endowed with the compact open topology is a topological abelian group, which will be called G ^/\ .

The group G is called reflexive if the natural embedding \alpha_G from G into the bidual G ^{/\ /\} : = (G ^/\ ) ^/\ 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 R^\omega or 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:

1. the ultrafilter { A subset X : x in A} is in \Xi(x),
2. if F in \Xi(x) and G in \Xi(x), then the filter F \cap G = { F \cup G : F in F, G in G } also belongs to \Xi(x),
3. if F in \Xi(x) and G contains F then G in \Xi(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 \GammaG the set of all continuous homomorphisms (in the sense of convergence) from G into T and the continuous convergence structure \Lambda, in \GammaG, is defined in the following way: a filter F in \GammaG converges in \Lambda to an element \xi in \GammaG if for every x in G and every filter H in G, convergent to x, w(F ×H) converges to \xi(x) in T. (F ×H denotes the filter generated by the products F ×G, F in F and H in H, and w(F ×H) : = { f(x); f in F and x in H }). Thus \Lambda is the coarsest convergence structure in \GammaG 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 \GammaG 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 \kappa_G : G --> \Gamma_c \Gamma_c 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. 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.

1 Banaszczyk, W. On the existence of Exotic Banach-Lie Groups Math. Ann. 264 1983 485-493

2 Banaszczyk, W. - Chasco, M.J. - Martín-Peinador, E. Open Subgroups and Pontryagin Duality Mathematische Zeitschrift 215 1994 195-204

3 Binz, E. Continuous Convergence in C(X) Lecture Notes in Mathematics 469. Springer-Verlag, Berlin Heidelberg New York, 1975

4 Butzmann, H.P. Pontrjagin-Dualität für topologische Vektorräume Arch. Math. 28 1977 632-637

5 Chasco, M.J. - Martín-Peinador, E. Binz-Butzmann duality versus Pontryagin Duality Archiv der Math. 63 No.3 1994 264-270

6 Fischer, H.R. Limesräume Math. Ann. 137 1959 269-303

7 Kaplan, S. Extension of the Pontryagin Duality. I: Infinite Products Duke Math. J. 15 1948 649-658

8 Martín-Peinador, E. A reflexive admissible topological group must be locally compact Proc. Amer. Math. Soc. 123 1995 3563-3566

9 Smith Freundlich, M. The Pontryagin duality theorem in linear spaces Ann. of Math. (2) 56 1952 248-253

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-10.