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Approximately transitive flows and ITPFI factors. (English) Zbl 0606.46041
The authors introduce a property of ergodic nonsingular group actions on a Lebesgue measure space that they call approximate transitivity. The main concern of the paper is with transformations and flows, but the definition of approximate transitivity is given for a Borel group G and a Borel homomorphism \(\alpha\) of G into the automorphism group of a Lebesgue measure space (x,\(\nu)\): \(\alpha\) is approximately transitive if, given \(n\in {\mathbb{N}}\), finite measures \(\mu_ 1,...,\mu_ n\prec \nu\), and \(\epsilon >0\), there exists a finite measure \(\mu\prec \nu\), and, for some \(m\in {\mathbb{N}}\), there are \(g_ 1,...,g_ m\in G\), and \(\lambda_{jk}\geq 0\) \((k=1,...,m\); \(j=1,...,n)\) such that \(\| \mu - \sum^{m}_{k=1}\lambda_{jk}\alpha_{g_ k}(\mu)\| <\epsilon\) (1\(\leq j\leq n)\). The motivation for introducing the notion of approximate transitivity is that the concept yields an answer to the problem of how the property of being ITPFI is reflected in the structure of the flow of weights of a factor. It is proved that a type \(III_ 0\) hyperfinite factor is ITPFI if and only if its flow of weights is approximately transitive, or, equivalently, that a type \(III_ 0\) ergodic nonsingular transformation has an orbit structure of infinite product type if and only if the associated flow is approximately transitive. Among others, the following facts are established for approximate transitivity:
A flow built over a base transformation with a constant ceiling function is approximately transitive if and only if the base transformation is approximately transitive.
Ergodic transformations and ergodic flows that have pure point spectrum are approximately transitive.
A finite measure-preserving approximately transitive transformation has zero entropy.

46L55 Noncommutative dynamical systems
46L35 Classifications of \(C^*\)-algebras
28D15 General groups of measure-preserving transformations
28D10 One-parameter continuous families of measure-preserving transformations
Full Text: DOI
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