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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.

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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