G-continuous functions and whirly actions

Abstract

This paper continues the work Glasner-Tsirelson-Weiss, ArXiv math.DS/0311450. For a Polish group G the notions of G-continuous functions and whirly actions are further exploited to show that: (i) A G-action is whirly iff it admits no nontrivial spatial (= pointwise) factors. (ii) Every action of a Polish Levy group is whirly. (iii) There exists a Polish monothetic group which is not Levy but admits a whirly action. (iv) In the Polish group Aut(X,μ), for the generic automorphism T, the action of the Polish group Λ(T) = closure {T^n: n \in Z} \subset Aut(X,μ) on the Lebesgue space (X,μ) is whirly. (v) The Polish additive group underlying a separable Hilbert space admits both spatial and whirly faithful actions. (vi) When G is a non-archimedean Polish group then every G-action is spatial.