Isomorphisms and non-isomorphisms of \(\mathrm{AT}\) actions
type: Item
instance of: scholarly article
title: Isomorphisms and non-isomorphisms of \(\mathrm{AT}\) actions
zbMATH Open document ID: 1196.60081
published in: Journal d'Analyse Mathématique
publication date: 2009-11-02T00:00:00Z
review text: The measure-theoretical classification of ergodic actions of \(\mathbb Z\) in terms of von Neumann algebras by \textit{W. Krieger} [Math. Ann. 223, 19--70 (1976; Zbl 0332.46045)] and by \textit{A. Connes} and \textit{E. J. Woods} [Pac. J. Math. 137, No. 2, 225--243 (1989; Zbl 0679.46051)], including the classification of approximately transitive (AT) actions by their Poisson boundary obtained as an inverse limit of polynomials, has recently been recast by \textit{T. Giordano} and the author [Münster J. Math. 1, No. 1, 15--72 (2008; Zbl 1163.60019)] without using von Neumann algebras or inverse limits. This gives a measure-theoretical version of dimension groups, together with a corresponding equivalence relation on diagrams. Here the programme is carried further, by developing computable invariants in terms of direct limits of binomial or truncated Poisson distributions. These extend and further develop the work of \textit{T. Giordano} and \textit{G. Skandalis} [J. Funct. Anal. 64, 209--226 (1985; Zbl 0586.46048)] on criteria for the pure point spectrum and of Connes and Woods [loc. cit.] on the so-called T-set, creating new invariants that are effective in some settings where the T-set is not. This lengthy and technically demanding paper is leavened by careful motivational and organisation asides and a judicious use of witty asides: sections entitled `The median is the message', `Beta behavior', Appendix E entitled `All about \(\mathcal E\)', a section titled `Weak isomorphism results' meaning weak results rather than results concerning weak isomorphism, and so on.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 28D20
zbMATH DE Number: 5624024
zbMATH Keywords: ergodic action
zbMATH Keywords: AT action
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: measure classification
author: David Handelman
reviewed by: Thomas B. Ward
MaRDI profile type: Publication
full work available at URL: s11854-009-0026-z
OpenAlex ID: W2086979838
cites work: Total variation asymptotics for sums of independent integer random variables
cites work: Hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks
cites work: Q3601976
cites work: Krieger factors isomorphic to their tensor square and pure point spectrum flows
cites work: Q3728664
cites work: Q3234855
cites work: Q4896037
cites work: An approximation theorem for the Poisson binomial distribution
cites work: Asymptotics and sharp bounds in the Poisson approximation to the Poisson-binomial distribution
DOI: 10.1007/S11854-009-0026-Z
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description: scientific article; zbMATH DE number 5624024