Spatial and Non-spatial Actions of Polish Groups

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 Aut (X,X, μ), for the generic automorphism T the action of the subgroup Λ(T ) = cls {Tn : n ∈ Z} on the Lebesgue space (X,X, μ) is whirly.