Eberlein oligomorphic groups

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 No-stable, No-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.