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Transactions of the American Mathematical Society

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Eberlein oligomorphic groups


Authors: Itaï Ben Yaacov, Tomás Ibarlucía and Todor Tsankov
Journal: Trans. Amer. Math. Soc. 370 (2018), 2181-2209
MSC (2010): Primary 22F50; Secondary 03C45, 22A25, 03C98, 22A15
DOI: https://doi.org/10.1090/tran/7227
Published electronically: November 28, 2017
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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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Additional Information

Itaï Ben Yaacov
Affiliation: Univ Lyon, Université Claude Bernard Lyon 1, Institut Camille Jordan, CNRS UMR 5208 43 boulevard du 11 novembre 1918 69622 Villeurbanne Cedex France

Tomás Ibarlucía
Affiliation: Univ Lyon, Université Claude Bernard Lyon 1, Institut Camille Jordan, CNRS UMR 5208 43 boulevard du 11 novembre 1918 69622 Villeurbanne Cedex France
Address at time of publication: Institut de Mathématiques de Jussieu–PRG Université Paris 7, case 7012 75205 Paris cedex 13 France
Email: ibarlucia@math.univ-paris-diderot.fr

Todor Tsankov
Affiliation: Institut de Mathématiques de Jussieu–PRG Université Paris 7, case 7012 75205 Paris cedex 13 France
Email: todor@math.univ-paris-diderot.fr

DOI: https://doi.org/10.1090/tran/7227
Keywords: Hilbert compactification, oligomorphic, $\aleph_0$-categorical, Fourier--Stieltjes algebra, semitopological semigroup compactification, inverse semigroup, Hilbert-representable
Received by editor(s): January 8, 2017
Published electronically: November 28, 2017
Additional Notes: The authors were partially supported by the ANR contract GrupoLoco (ANR-11-JS01-0008). The second author was partially supported by the ANR contract ValCoMo (ANR-13-BS01-0006). The third author was partially supported by the ANR contract GAMME (ANR-14-CE25-0004).
Article copyright: © Copyright 2017 American Mathematical Society

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