The concept of boundedness and the Bohr compactification of a MAP Abelian group

Jorge Galindo¹ ; Salvador Hernández¹

¹

Fundamenta Mathematicae, Tome 159 (1999) no. 3, pp. 195-218.

Résumé

Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an $ℒ_∞$-group, contains "many" discrete C-embedded subsets which are C*-embedded in their Bohr compactification. This result generalizes an analogous theorem of van Douwen for the discrete case and some other ones due to Hartman and Ryll-Nardzewski concerning the existence of $I_0$-sets.  We also obtain some results on preservation of compactness for the Bohr topology of several types of MAP Abelian groups, like $ℒ_∞$-groups, locally convex vector spaces and free Abelian topological groups.

Détail

DOI : 10.4064/fm-159-3-195-218

Keywords: Bohr topology, LCA group, $ℒ_∞$-group, boundedness, locally convex vector space, DF-space, maximally almost periodic, respects compactness, C-embedded, C*-embedded