G-continuous Functions and Whirly Actions

This paper continues the work [6]. 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 factors. (ii) Every action of a Polish Lévy group is whirly. (iii) There exists a Polish monothetic group which is not Lévy but admits a whirly action. (iv) In the Polish group Aut (X,X, μ), for the generic automorphism T the action of the Polish group Λ(T ) = cls {T n : n ∈ Z} ⊂ Aut (X) on the Lebesgue space (X,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. In the work [6] the authors established, given a boolean action of a Polish group G on a measure algebra (X, μ), a necessary and sufficient condition for the action (X, μ, G) to admit a spatial (or a pointwise) model. This necessary and sufficient condition was formulated in terms of certain functions in L(μ) called G-continuous functions. Another key notion introduced in [6] was that of a whirly G-action. It was shown there that a whirly action admits no nontrivial spatial factors. In the present work, which is a natural continuation of [6], we exploit these new notions and results in several ways. In the first introductory section we recall the relevant facts from [6]. In the second section we define stable sets and show that the sub σ-algebra of X generated by the G-continuous functions coincides with the σ-algebra generated by the stable sets. As corollaries we deduce that: (i) A G-action is whirly iff it admits no nontrivial spatial factors. (ii) Every action of a Polish Lévy group is whirly. In Section three we deduce some facts concerning the structure of whirly and spatial systems. In Section four we show that for a non-archimedean Polish group every boolean action is spatial. In the fifth section we show that in the Polish group Aut (X,X, μ) of automorphisms of a standard Lebesgue space, (topologically) almost every automorphism defines a whirly action on (X,X, μ). In the final section we show that there are Polish groups which admit a whirly action and yet are not Lévy. First we find T ∈ Aut (X) for which Λ(T ) has this property, and then we show that the abelian Polish group which underlies a separable infinite dimensional Hilbert space H admits both spatial and whirly faithful actions. We thank Matt Foreman for a helpful conversation.