Left and right uniform structures on functionally balanced groups
by
Ahmed Bouziad
Universite de Rouen, France
Coauthors: Jean-Pierre Troallic, Universite du Havre, France

Let G be a Hausdorff topological group. It is shown that there is a class C of subspaces of G, containing all (but not only) precompact subsets of G, for which the following result holds:

Suppose that for every real-valued discontinuous function on G there is a set A in C such that the restriction mapping f_|A has no continuous extension to G; then the following are equivalent:

(i) the left and right uniform structures of G are equivalent, (ii) every left uniformly continuous bounded real-valued function on G is right uniformly continuous, (iii) for every countable set A subset G and every neighborhood U of the unit e of G, there is a neighborhood V of e in G such that AV subset UA.

As a consequence, it is proved that items (i), (ii) and (iii) are equivalent for every inframetrizable group. These results generalize earlier ones established by Itzkowitz, Rothman, Strassberg and Wu, by Milnes and by Pestov for locally compact groups, by Protasov for almost metrizable groups, and by Troallic for groups that are quasi-k-spaces.

Date received: May 15, 2003