Topological annihilators in compact abelian groups
by
Gábor Lukács
Dalhousie University, Halifax, Nova Scotia, Canada

The Pontryagin dual of an abelian topological group G is the group [^G]=\mathscrH(G, T) of all continuous characters of G equipped with the compact-open topology (T is the circle group). Following Dikranjan, Milan and Tonolo [2.1, 5], for a sequence u=(u_n) in [^G], we put s_u(G)={x Î G | u_n(x)® 0 in T}, the subgroup of topologically u-annihilating elements of G. For every subgroup H £ G, we set

gG(H)= Ç { su(G) | u Î ^ G   N   , H £ su(G)}.

(1)

One says that H £ G is g-closed in G if g_G(H)=H, and H is g-dense there if g_G(H)=G. Using this terminology, g is the largest closure operator with respect to which each basic g-closed set s_u(G) is closed. (For details on closure operators of topological groups, we refer the reader to the survey of Dikranjan [3].)

In this talk, we focus on the action of g on subgroups of compact Hausdorff abelian groups. Given a subgroup H of a compact Hausdorff abelian group K, H defines a group topology t_H on the discrete group A=[^K] (namely, the topology of pointwise convergence on H). Our approach is to understand the closure operator g by studying the functor (A, t_H)® (A, t_g(H)). Two of our main results are the following theorems:

Theorem 1 Let H be a countable subgroup of a compact Hausdorff abelian topological group K. Then H is g-closed in K.

Theorem 1 is a positive solution for two problems of Dikranjan, Milan and Tonolo [Problem 5.1, Question 5.2, 5], and it generalizes a result of Bíró, Deshouillers and Sós [Theorem 2, 2]. (Independently, Dikranjan and Kunen [4] and Mathias Beiglböck, Christian Steineder and Reinhard Winkler [1] have also obtained Theorem 1.)

Theorem 2 Every g-closed subgroup of a compact Hausdorff abelian group is realcompact.

If the time will permit, a "relative" of g will also be discussed, which turns out to coincide with the G_d-closure (and thus the Hewitt-realcompactification) on compact Hausdorff groups.

Bibliography

1. Mathias Beiglböck, Christian Steineder, and Reinhard Winkler. Sequences and filters of characters characterizing subgroups of compact abelian groups. Preprint, ArXiv: math.GN/0412447.

2. A. Bíró, J.-M. Deshouillers, and V. T. Sós. Good approximation and characterization of subgroups of R/Z. Studia Sci. Math. Hungar., 38:97-113, 2001.

3. Dikran Dikranjan. Closure operators in topological groups related to von neumann's kernel. Topology Appl., (to appear).

4. Dikran Dikranjan and Kenneth Kunen. Characterizing subgroups of compact abelian groups. Preprint, ArXiv: math.GN/0412327.

5. Dikran Dikranjan, Chiara Milan, and Alberto Tonolo. A characterization of the maximally almost periodic abelian groups. J. Pure Appl. Algebra, 197(1-3):23-41, 2005.

Date received: June 2, 2005