Submission date: 16 February 2016.

Abstract:

We study the Fourier-Stieltjes algebra of Roelcke precompact, non-archimedean, Polish groups and give a model-theoretic description of the Hilbert compactification of these groups. We characterize the family of such groups whose Fourier-Stieltjes algebra is dense in the algebra of weakly almost periodic functions: those are exactly the automorphism groups of ℵ_0-stable, ℵ_0-categorical structures. This analysis is then extended to all semitopological semigroup compactifications S of such a group: S is Hilbert-representable if and only if it is an inverse semigroup.
We also show that every factor of the Hilbert compactification is Hilbert-representable.

Mathematics Subject Classification:

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Full text arXiv 1602.05097: pdf, ps.