Spatial and non-spatial actions of Polish groups.

*(English)*Zbl 1085.54027As 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.

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.

Reviewer: Badri P. Dvalishvili (Tbilisi)

##### MSC:

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