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Spatial and non-spatial actions of Polish groups

Published online by Cambridge University Press:  04 August 2005

E. GLASNER
Affiliation:
Department of Mathematics, Tel Aviv University, Ramat Aviv, Israel (e-mail: glasner@math.tau.ac.il)
B. WEISS
Affiliation:
Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel (e-mail: weiss@math.huji.ac.il)
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Abstract

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For locally compact groups all actions on a standard measure algebra have a spatial realization. For many Polish groups this is no longer the case. However, we show here that for non-archimedean Polish groups all measure algebra actions do have spatial realizations. In the other direction we show that an action of a Polish group is whirly (‘ergodic at the identity’) if and only if it admits no spatial factors and that all actions of a Lévy group are whirly. We also show that in the Polish group $\rm{Aut\,}(X,\mathcal{X},\mu)$, for the generic automorphism T the action of the subgroup $\Lambda(T)=\rm{cls\,}\{T^n: n\in \mathbb{Z}\}$ on the Lebesgue space $(X,\mathcal{X},\mu)$ is whirly.

Information

Type
Research Article
Copyright
2005 Cambridge University Press