Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [AG14] Ethan Akin and Eli Glasner, WAP Systems and Labeled Subshifts, arXiv:1410.4753 [math.DS].
  • [AZ86] Gisela Ahlbrandt and Martin Ziegler, Quasi-finitely axiomatizable totally categorical theories, Ann. Pure Appl. Logic 30 (1986), no. 1, 63-82. Stability in model theory (Trento, 1984). MR 831437, https://doi.org/10.1016/0168-0072(86)90037-0
  • [BBH14] Itaï Ben Yaacov, Alexander Berenstein, and C. Ward Henson, Almost indiscernible sequences and convergence of canonical bases, J. Symb. Log. 79 (2014), no. 2, 460-484. MR 3224976, https://doi.org/10.1017/jsl.2013.38
  • [BJM78] J. F. Berglund, H. D. Junghenn, and P. Milnes, Compact right topological semigroups and generalizations of almost periodicity, Lecture Notes in Mathematics, vol. 663, Springer, Berlin, 1978. MR 513591
  • [BLM01] A. Bouziad, M. Lemańczyk, and M. K. Mentzen, A compact monothetic semitopological semigroup whose set of idempotents is not closed, Semigroup Forum 62 (2001), no. 1, 98-102. MR 1832257, https://doi.org/10.1007/s002330010016
  • [BT16] Itaï Ben Yaacov and Todor Tsankov, Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups, Trans. Amer. Math. Soc. 368 (2016), no. 11, 8267-8294. MR 3546800, https://doi.org/10.1090/tran/6883
  • [Cho82] Ching Chou, Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups, Trans. Amer. Math. Soc. 274 (1982), no. 1, 141-157. MR 670924, https://doi.org/10.2307/1999501
  • [Ebe49] W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217-240. MR 0036455, https://doi.org/10.2307/1990424
  • [Fol95] Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
  • [Gla12] Eli Glasner, The group $ {\rm Aut}(\mu)$ is Roelcke precompact, Canad. Math. Bull. 55 (2012), no. 2, 297-302. MR 2957245, https://doi.org/10.4153/CMB-2011-083-2
  • [GM06] E. Glasner and M. Megrelishvili, Hereditarily non-sensitive dynamical systems and linear representations, Colloq. Math. 104 (2006), no. 2, 223-283. MR 2197078, https://doi.org/10.4064/cm104-2-5
  • [GM08] E. Glasner and M. Megrelishvili, New algebras of functions on topological groups arising from $ G$-spaces, Fund. Math. 201 (2008), no. 1, 1-51. MR 2439022, https://doi.org/10.4064/fm201-1-1
  • [GM14] Eli Glasner and Michael Megrelishvili, Representations of dynamical systems on Banach spaces, Recent progress in general topology. III, Atlantis Press, Paris, 2014, pp. 399-470. MR 3205489, https://doi.org/10.2991/978-94-6239-024-9_9
  • [Gro52] A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168-186 (French). MR 0047313, https://doi.org/10.2307/2372076
  • [GW12] Eli Glasner and Benjamin Weiss, On Hilbert dynamical systems, Ergodic Theory Dynam. Systems 32 (2012), no. 2, 629-642. MR 2901363, https://doi.org/10.1017/S0143385711000277
  • [How95] John M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs. New Series, vol. 12, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1455373
  • [HR70] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
  • [Iba16] Tomás Ibarlucía, The dynamical hierarchy for Roelcke precompact Polish groups, Israel J. Math. 215 (2016), no. 2, 965-1009. MR 3552300, https://doi.org/10.1007/s11856-016-1399-1
  • [Kim14] Byunghan Kim, Simplicity theory, Oxford Logic Guides, vol. 53, Oxford University Press, Oxford, 2014. MR 3156332
  • [Law74] J. D. Lawson, Joint continuity in semitopological semigroups, Illinois J. Math. 18 (1974), 275-285. MR 0335674
  • [Law84] Jimmie D. Lawson, Points of continuity for semigroup actions, Trans. Amer. Math. Soc. 284 (1984), no. 1, 183-202. MR 742420, https://doi.org/10.2307/1999282
  • [Meg01] Michael G. Megrelishvili, Operator topologies and reflexive representability, Nuclear groups and Lie groups (Madrid, 1999) Res. Exp. Math., vol. 24, Heldermann, Lemgo, 2001, pp. 197-208. MR 1858149
  • [Meg03] Michael Megrelishvili, Fragmentability and representations of flows, Proceedings of the 17th Summer Conference on Topology and its Applications, 2003, pp. 497-544. MR 2077804
  • [Meg07] Elliott Pearl, Open problems in topology, Topology Appl. 136 (2004), no. 1-3, 37-85. MR 2023411, https://doi.org/10.1016/S0166-8641(03)00183-4
  • [Meg08] Michael Megrelishvili, Reflexively representable but not Hilbert representable compact flows and semitopological semigroups, Colloq. Math. 110 (2008), no. 2, 383-407. MR 2353912, https://doi.org/10.4064/cm110-2-5
  • [Pil96] Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
  • [Rud59] Walter Rudin, Weak almost periodic functions and Fourier-Stieltjes transforms, Duke Math. J. 26 (1959), 215-220. MR 0102705
  • [Sht94] Alexander I. Shtern, Compact semitopological semigroups and reflexive representability of topological groups, Russian J. Math. Phys. 2 (1994), no. 1, 131-132. MR 1297947
  • [Tsa12] Todor Tsankov, Unitary representations of oligomorphic groups, Geom. Funct. Anal. 22 (2012), no. 2, 528-555. MR 2929072, https://doi.org/10.1007/s00039-012-0156-9
  • [TZ12] Katrin Tent and Martin Ziegler, A course in model theory, Lecture Notes in Logic, vol. 40, Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012. MR 2908005
  • [Usp98] V. V. Uspenskij, The Roelcke compactification of unitary groups, Abelian groups, module theory, and topology (Padua, 1997) Lecture Notes in Pure and Appl. Math., vol. 201, Dekker, New York, 1998, pp. 411-419. MR 1651186
  • [Vee79] William A. Veech, Weakly almost periodic functions on semisimple Lie groups, Monatsh. Math. 88 (1979), no. 1, 55-68. MR 550072, https://doi.org/10.1007/BF01305857
  • [Wag94] Frank O. Wagner, Relational structures and dimensions, Automorphisms of first-order structures, Oxford Sci. Publ., Oxford Univ. Press, New York, 1994, pp. 153-180. MR 1325473

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 22F50, 03C45, 22A25, 03C98, 22A15

Retrieve articles in all journals with MSC (2010): 22F50, 03C45, 22A25, 03C98, 22A15


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

American Mathematical Society