Eberlein oligomorphic groups

Itaï Ben Yaacov (1) , Tomás Ibarlucía (1, 2, 3) , Todor Tsankov (4)

1 AGL - Algèbre, géométrie, logique
2 UCBL - Université Claude Bernard Lyon 1
3 ICJ - Institut Camille Jordan
4 IMJ - Institut de Mathématiques de Jussieu

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 compactifica-tions 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.

Mots clés: semitopological semigroup compactification; Hilbert-representable systems; Fourier–Stieltjes algebra; Hilbert compactification; inverse semigroup; $aleph_0$-categorical; oligomorphic

Domaines: Logique [math.LO]; Systèmes dynamiques [math.DS]

Itaï Ben Yaacov, Tomás Ibarlucía, Todor Tsankov. Eberlein oligomorphic groups. Transactions of the American Mathematical Society, 2017, 370 (3), pp.2181-2209. ⟨10.1090/tran/7227⟩. ⟨hal-01275112v3⟩