Left and right uniform structures on functionally balanced groups |
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Authors: | A Bouziad JP Troallic |
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Institution: | a Département de Mathématiques, Université de Rouen, UMR CNRS 6085, F-76821 Mont Saint Aignan, France b Faculté des Sciences et Techniques, Université du Havre, 25 rue P. Lebon, F-76600 Le Havre, France |
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Abstract: | 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∈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⊂G and every neighborhood V of the unit e of G, there is a neighborhood U of e in G such that AU⊂VA.
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. |
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Keywords: | 22A05 54E15 |
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