Spatial and non-spatial actions of Polish groups

Abstract

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.