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Spatial models of Boolean actions and groups of isometries

Published online by Cambridge University Press:  11 February 2010

ALEKSANDRA KWIATKOWSKA  and
SŁAWOMIR SOLECKI
ALEKSANDRA KWIATKOWSKA
Affiliation:
Department of Mathematics, 1409 W. Green Street, University of Illinois, Urbana, IL 61801, USA (email: ssolecki@illinois.edu)
SŁAWOMIR SOLECKI
Affiliation:
Department of Mathematics, 1409 W. Green Street, University of Illinois, Urbana, IL 61801, USA (email: ssolecki@illinois.edu)
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Abstract

Given a Polish group G of isometries of a locally compact separable metric space, we prove that each measure-preserving Boolean action by G has a spatial model or, in other words, has a point realization. This result extends both a classical theorem of Mackey and a recent theorem of Glasner and Weiss, and it covers interesting new examples. In order to prove our result, we give a characterization of Polish groups of isometries of locally compact separable metric spaces which may be of independent interest. The solution to Hilbert’s fifth problem plays an important role in establishing this characterization.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 2 , April 2011 , pp. 405 - 421
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Aliprantis, C. D. and Burkinshaw, O.. Principles of Real Analysis. Academic Press, New York, NY, 1998.Google Scholar
[2]Becker, H.. A Boolean G-space without a point realization. Manuscript, December 2001.Google Scholar
[3]Conway, J. B.. A Course in Functional Analysis. Springer, Berlin, 1990.Google Scholar
[4]Danilenko, A. I.. Point realization of Boolean actions of countable inductive limits of locally compact groups. Mat. Fiz. Anal. Geom. 7 (2000), 35–48.Google Scholar
[5]Gao, S.. Invariant Descriptive Set Theory. CRC Press, Boca Raton, FL, 2009.Google Scholar
[6]Gao, S. and Kechris, A. S.. On the classification of Polish metric spaces up to isometry. Mem. Amer. Math. Soc. 161(766) (2003).Google Scholar
[7]Glasner, E., Tsirelson, B. and Weiss, B.. The automorphism group of the Gaussian measure cannot act pointwise. Israel J. Math. 148 (2005), 305–329.CrossRefGoogle Scholar
[8]Glasner, E. and Weiss, B.. Spatial and non-spatial actions of Polish groups. Ergod. Th. & Dynam. Sys. 25 (2005), 1521–1538.CrossRefGoogle Scholar
[9]Mackey, G. W.. Point realisations of transformation groups. Illinois J. Math. 6 (1962), 327–335.CrossRefGoogle Scholar
[10]Malicki, M. and Solecki, S.. Isometry groups of separable metric spaces. Math. Proc. Cambridge Philos. Soc. 146 (2009), 67–81.CrossRefGoogle Scholar
[11]Melleray, J.. Compact metrizable groups are isometry groups of compact metric spaces. Proc. Amer. Math. Soc. 136 (2008), 1451–1455.CrossRefGoogle Scholar
[12]Montgomery, D. and Zippin, L.. Topological Transformation Groups. Interscience Publishers, New York, 1955.Google Scholar
[13]Uspenskij, V. V.. On the group of isometries of the Urysohn universal metric space. Comment. Math. Univ. Carolin. 31 (1990), 181–182.Google Scholar
[14]Warner, F. W.. Foundations of Differentiable Manifolds and Lie Groups. Springer, Berlin, 1983.CrossRefGoogle Scholar