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

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 I0-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.