Isomorphisms and non-isomorphisms of \(\mathrm{AT}\) actions
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.
scientific article; zbMATH DE number 5624024
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