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The automorphism group of the Gaussian measure cannot act pointwise. (English) Zbl 1105.37006

The authors prove that a theorem of Mackey saying that every Boolean \(G\)-action (i.e., a near action of \(G\)) admits a spatial model, does not hold for actions of Polish groups. The paper is organized as follows. Using ideas from M. Gromov and V. D. Milman [Am. J. Math. 105, 843–854 (1983; Zbl 0522.53039)], they first show that a Lévy group (defined as a Polish group with additional properties) admits only trivial spacial actions. Later, they give a necessary and sufficient condition for a near action to admit a spacial model in terms of \(G\)-continuous functions. Finally, they give a second proof of the absence of a spacial model by proving that the near-action of the automorphism group of the Gaussian measure is whirly.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54H20 Topological dynamics (MSC2010)
37A15 General groups of measure-preserving transformations and dynamical systems
28D15 General groups of measure-preserving transformations

Citations:

Zbl 0522.53039
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References:

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