Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-26T08:37:36.470Z Has data issue: false hasContentIssue false
  • English
  • Français

AUTOMORPHISM GROUPS OF RANDOMIZED STRUCTURES

Published online by Cambridge University Press:  08 September 2017

TOMÁS IBARLUCÍA
TOMÁS IBARLUCÍA*
Affiliation:
UNIVERSITÉ DE LYON INSTITUT CAMILLE JORDAN 43 BLVD. DU 11 NOVEMBRE 1918 69622 VILLEURBANNE CEDEX FRANCE E-mail: ibarlucia@math.univ-lyon1.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We study automorphism groups of randomizations of separable structures, with focus on the ℵ0-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the ℵ0-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and classify the separable models of the theory of beautiful pairs of randomizations, showing in particular that this theory is never ℵ0-categorical (except in basic cases).

Keywords

Primary 22F5022F1022A0503C30Secondary 22A1503C45randomizationmeasurable wreath productℵ0-categoricalRoelcke precompactbeautiful pairs
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 3 , September 2017 , pp. 1150 - 1179
Copyright
Copyright © The Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, U., Goldbring, I., and Keisler, H. J., Definable closure in randomizations . Annals of Pure and Applied Logic, vol. 166 (2015), no. 3, pp. 325–341.CrossRefGoogle Scholar
Andrews, U., Goldbring, I., and Keisler, H. J., Independence relations in randomizations, arXiv:1409.1531 [math.LO].Google Scholar
Andrews, U. and Keisler, H. J., Separable models of randomizations, this Journal, vol. 80 (2015), no. 4, pp. 1149–1181.Google Scholar
Ben Yaacov, I., Continuous and random Vapnik-Chervonenkis classes . Israel Journal of Mathematics, vol. 173 (2009), pp. 309–333.Google Scholar
Ben Yaacov, I., On uniform canonical bases in Lp lattices and other metric structures . Journal of Logic and Analysis, vol. 4 (2012), Paper 12, 30.Google Scholar
Ben Yaacov, I., Model theoretic stability and definability of types, after A. Grothendieck . Bulletin of Symbolic Logic, vol. 20 (2014), no. 4, pp. 491–496.Google Scholar
Ben Yaacov, I., On theories of random variables . Israel Journal of Mathematics, vol. 194 (2013), no. 2, pp. 957–1012.Google Scholar
Ben Yaacov, I., On Roelcke precompact Polish groups which cannot act transitively on a complete metric space , to appear in Israel Journal of Mathematics, arXiv:1510.00238 [math.LO].Google Scholar
Ben Yaacov, I., Berenstein, A., and Henson, C. W., Almost indiscernible sequences and convergence of canonical bases, this Journal, vol. 79 (2014), no. 2, pp. 460–484.Google Scholar
Ben Yaacov, I., Berenstein, A., Henson, C. W., and Usvyatsov, A., Model theory for metric structures , Model Theory with Applications to Algebra and Analysis, vol. 2 (Chatzidakis, Z., Macpherson, D., Pillay, A., Wilkie, A., editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 315–427.Google Scholar
Ben Yaacov, I., Ibarlucía, T., and Tsankov, T., Eberlein oligomorphic groups, arXiv:1602.05097 [math.LO].Google Scholar
Ben Yaacov, I. and Keisler, H. J., Randomizations of models as metric structures . Confluentes Mathematici, vol. 1 (2009), no. 2, pp. 197–223.CrossRefGoogle Scholar
Ben Yaacov, I. and Tsankov, T., Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups . Transactions of the American Mathematical Society, vol. 368 (2016), no. 11, pp. 8267–8294.CrossRefGoogle Scholar
Ben Yaacov, I. and Usvyatsov, A., On d-finiteness in continuous structures . Fundamenta Mathematicae, vol. 194 (2007), no. 1, pp. 67–88.Google Scholar
Ben Yaacov, I. and Usvyatsov, A., Continuous first order logic and local stability . Transactions of the American Mathematical Society, vol. 362 (2010), no. 10, pp. 5213–5259.Google Scholar
Berberian, S. K., Lectures in Functional Analysis and Operator Theory, Graduate Texts in Mathematics, vol. 15, Springer-Verlag, New York-Heidelberg, 1974.CrossRefGoogle Scholar
Cembranos, P. and Mendoza, J., Banach Spaces of Vector-Valued Functions, Lecture Notes in Mathematics, vol. 1676, Springer-Verlag, Berlin, 1997.Google Scholar
Diestel, J. and Uhl, J. J. Jr., Vector Measures , American Mathematical Society, Providence, RI, 1977, With a foreword by Pettis, B. J., Mathematical Surveys, No. 15.Google Scholar
Fremlin, D. H., Measure Theory, vol. 4, Torres Fremlin, Colchester, 2006, Topological measure spaces. Part I, II, Corrected second printing of the 2003 original.Google Scholar
Glasner, E., Ergodic Theory via Joinings, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003.Google Scholar
Glasner, E. and Megrelishvili, M., Representations of dynamical systems on Banach spaces not containing l 1 . Transactions of the American Mathematical Society, vol. 364 (2012), no. 12, pp. 6395–6424.Google Scholar
Glasner, E. and Megrelishvili, M., Representations of Dynamical Systems on Banach Spaces, Recent Progress in General Topology, III, Atlantis Press, Paris, 2014, pp. 399–470.Google Scholar
Ibarlucía, T., The dynamical hierarchy for Roelcke precompact Polish groups . Israel Journal of Mathematics, vol. 215 (2016), no. 2, pp. 965–1009.Google Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.Google Scholar
Kechris, A. S., Global Aspects of Ergodic Group Actions, Mathematical Surveys and Monographs, vol. 160, American Mathematical Society, Providence, RI, 2010.CrossRefGoogle Scholar
Keisler, H. J., Randomizing a model . Advances in Mathematics, vol. 143 (1999), no. 1, pp. 124–158.Google Scholar
Kaïchouh, A. and Le Maître, F., Connected Polish groups with ample generics . Bulletin of the London Mathematical Society, vol. 47 (2015), no. 6, 996–1009.CrossRefGoogle Scholar
Le Maître, F., Sur les groupes pleins préservant une mesure de probabilité, Ph.D. thesis, École Normale Supérieure de Lyon, 2014.Google Scholar
Megrelishvili, M., Reflexively representable but not Hilbert representable compact flows and semitopological semigroups . Colloquium Mathematicum, vol. 110 (2008), no. 2, pp. 383–407.Google Scholar
Pillay, A., Geometric Stability Theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, Oxford Science Publications, New York, 1996.Google Scholar
Pisier, G., Une propriété de stabilité de la classe des espaces ne contenant pas l 1 . Comptes Rendus de l’Académie des Sciences. Series A-B, vol. 286 (1978), no. 17, pp. A747–A749.Google Scholar
Shtern, A. I., Compact semitopological semigroups and reflexive representability of topological groups . Russian Journal of Mathematical Physics, vol. 2 (1994), no. 1, pp. 131–132.Google Scholar
Tsankov, T., Unitary representations of oligomorphic groups . Geometric and Functional Analysis, vol. 22 (2012), no. 2, pp. 528–555.Google Scholar