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Spatial and non-spatial actions of Polish groups. (English) Zbl 1085.54027
As the classical theorems of G. W. Mackey and A. Ramsay showed, all measure algebra actions of a locally compact second countable topological group admit spatial realizations [C. W. Mackey, Ill. J. Math. 6, 327–335 (1962; Zbl 0178.17203) and A.Ramsey, Adv. Math. 6, 253–322 (1971; Zbl 0216.14902)]. In [Isr. J. Math. 148, 305–329 (2003; Zbl 1105.37006)], the authors of the paper under review jointly with B. Tsirelson have shown that this is no longer the case for a general Polish topological group.
However, in the second section of the paper the authors prove that for a non-Archimedian Polish group every near-action (or Boolean action) admits a spatial model. On the other hand, using the notion of a Levy group introduced and studied by M. Gromov and V. D. Milman [Am. J. Math. 105, 843–854 (1983; Zbl 0522.53039)] it was shown in the above-mentioned paper, not published for the present, that many Polish groups have near-actions which do not admit spatial models; in particular, it is shown that for a Polish Levy group any spatial action is necessarily trivial.
Furthermore, for a measure algebra \((\chi,\mu)\) the authors define stable sets and show that the sub-\(\sigma\)-algebra of \(\chi\) generated by the \(G\)-continuous functions (introduced by the same authors) coincides with the sub-\(\sigma\)-algebra generated by the stable sets. From this fact it easily follows that: (i) an action of \(G\) on a measure algebra \((\chi,\mu)\) is whirly, i.e. it is ergodic at the identity (\(\Longleftrightarrow\mu(UA)=1\) for every positive set \(A\in \chi\) and every neighborhood \(u\) of the identity \(e\in G\)) iff it admits no non-trivial spatial factors; (ii) every ergodic Boolean action of a Polish Levy group is whirly.
In the fourth and fifth section, the structure of whirly and spatial systems is studied together with the proof that in the Polish group \(\text{ Aut}\,(X,\chi,\mu)\) of automorphisms of a standard Lebesgue space, for the generic automorphism \(T\) the action of the subgroup \(\Lambda(T)=\text{ cls}(T^n:\;n\in \mathbb{Z})\) on the Lebesgue space \((X,\chi,\mu)\) is whirly. In the final, sixth section, the authors show that there are Polish groups which admit whirly actions, but not Levy’s ones.

MSC:
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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