Two observations on irreducible representations of groups.
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 -group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification. This...
Embedding a topological group into its WAP-compactification
We prove that the topology of the additive group of the Banach space c₀ is not induced by weakly almost periodic functions or, what is the same, that this group cannot be represented as a group of isometries of a reflexive Banach space. We show, in contrast, that additive groups of Schwartz locally convex spaces are always representable as groups of isometries on some reflexive Banach space.
On a theorem of van Douwen.
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