Ergodicity and type of nonsingular Bernoulli actions
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Ergodicity and type of nonsingular Bernoulli actions Michael Björklund1 · Zemer Kosloff2 · Stefaan Vaes3
Received: 27 February 2019 / Accepted: 15 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We determine the Krieger type of nonsingular Bernoulli actions G g∈G ({0, 1}, μg ). When G is abelian, we do this for arbitrary marginal measures μg . We prove in particular that the action is never of type II∞ if G is abelian and not locally finite, answering Krengel’s question for G = Z. When G is locally finite, we prove that type II∞ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from 0 and 1. When G has only one end, we prove that the Krieger type is always I, II1 or III1 . When G has more than one end, we show that other types always arise. Finally, we solve the conjecture of Vaes and Wahl (Geom Funct Anal 28:518–562, 2018)
M.B. was supported by GoCas Young Excellence Grant 11423310. Z.K. was partially supported by ISF Grant No. 1570/17. S.V. was supported by European Research Council Consolidator Grant 614195 RIGIDITY, and by long term structural funding—Methusalem grant of the Flemish Government.
B Stefaan Vaes
[email protected] Michael Björklund [email protected] Zemer Kosloff [email protected]
1
Department of Mathematics, Chalmers, Gothenburg, Sweden
2
Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
3
Department of Mathematics, KU Leuven, Leuven, Belgium
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by proving that a group G admits a Bernoulli action of type III1 if and only if G has nontrivial first L 2 -cohomology.
1 Introduction The Bernoulli actions G (X, μ) = ({0, 1}, μ0 )G of a countable group G, given by (g · x)h = x g−1 h , play a key role in ergodic theory, measurable group theory and operator algebras. By construction, μ is a G-invariant probability measure. Replacing μ by an arbitrary product measure μ = g∈G μg , one obtains a very natural family of non measure preserving G-actions. Although this construction is straightforward, it turned out to be a very difficult problem to decide when G (X, μ) is ergodic and, in that case, to determine the Krieger type of the action. The first results in this direction were providing examples for the group G = Z, through inductive constructions of probability measures (μn )n∈Z on {0, 1}. It was thus proven in [17] that there exists an ergodic Bernoulli shift without equivalent invariant probability measure, while in [11], it was shown that there are ergodic Bernoulli shifts of type III, i.e. without equivalent σ finite invariant measure. Finally, the first examples of Bernoulli shifts of type III1 were constructed in [14]. Proving ergodicity and determining the type of a nonsingular Bernoulli action is a difficult problem, because these actions may very well be dissipative (i.e. admit a fundamental domain), which was already proven in [11]. A first general result was obtained in [15] for G = Z and marginal measures (μn )n∈Z satis
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