Eberlein oligomorphic groups

Ben Yaacov, Itaï; Ibarlucía, Tomás; Tsankov, Todor

doi:10.1090/tran/7227

Eberlein oligomorphic groups
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by Itaï Ben Yaacov, Tomás Ibarlucía and Todor Tsankov PDF

Trans. Amer. Math. Soc. 370 (2018), 2181-2209 Request permission

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 $\aleph _0$-stable, $\aleph _0$-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.

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