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Approximation for actions of amenable groups and transversal automorphisms. (English) Zbl 0584.28008

Operator algebras and their connections with topology and ergodic theory, Proc. Conf., Buşteni/Rom. 1983, Lect. Notes Math. 1132, 331-346 (1985).
[For the entire collection see Zbl 0562.00005.]
For any measure-preserving action (m.p.) of \(G={\mathbb{Z}}\) there exists, as it was shown by the second author [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 115, 72-82 (1982; Zbl 0505.47006)], a so-called adic representation by odometer-like transformations of a (nonstationary) Markov compactum, the latter being a subset in some infinite product space consisting of sequences \((x_ i)\) with restrictions imposed on pairs \((x_ i,x_{i+1})\). This representation makes more precise the idea of a sequence of towers giving cyclic approximations of an automorphism.
An attempt to pass to the case when G is discrete amenable has lead the authors to the notions of transversal and generalized transversal \((=\) approximately cylindrical) transformations of a Markov compactum which generalize the notion of adic transformation and are connected with two ways of orderings of the compactum.
The first result of the paper is a criterion of a possibility to represent an individual m.p. transformation from G as a transversal one, i.e. to introduce a Markov compactum structure on the measure space, in such a way that an orbit partition of G becomes the tail partition. The second theorem shows that a m.p. action of an amenable group always admits a representation by generalized transversal transformations of a common Markov compactum.

MSC:

28D15 General groups of measure-preserving transformations
47A35 Ergodic theory of linear operators
46L55 Noncommutative dynamical systems