Isomorphisms and non-isomorphisms of at actions

Abstract

Measure-theoretic classification of ergodic actions of the integers and reals was developed through the use of von Neumann algebras by Krieger and later by Connes and Woods. In particular, the latter showed that AT (approximatively transitive) actions are classified by their Poisson boundary obtained from an inverse limit of (binomial) polynomials. Recently (2008), Giordano and Handelman [GH] showed that the classification can be divorced from the von Neumann algebra and inverse limit constructions, by instead examining (what amounts to the predual) direct limits. This results in a measure-theoretic version of dimension groups (used in C*-algebras, especially in classification programmes), and a corresponding equivalence relation on diagrams which amounts to determining isomorphism classes. The next step is to translate the equivalence relation to easily (or relatively easily) computable criteria, usually numerical, and this is the thrust of this paper. An ergodic action whose corresponding factor is said to have pure point spectrum if it is isomorphic to its tensor square, or equivalently, the ergodic action satisfies the analogous property. Giordano and Skandalis obtained a numerical sufficient condition in [GS]. Here expressed in terms of a direct limit of either binomial or truncated Poisson distributions (actually more general distributions, using variance), we obtain numerical conditions that are an improvement on those of [GS] in the cases in which the latter apply, and extend them to a much wider class, called relatively absorbing. Some surprises occur, particularly when the gaps in the corresponding random walk increase unboundedly. We also deal with invariants for isomorphism in the measure-theoretic dimension group setting. An old one, developed in [CW], is the T-set. We extend the idea to create new invariants that are effective when the T-set is not; again, these involve numerical data (involving computations of \( l^{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} } \), that is, total variation, distances) that are fairly tractible. We also give effective criteria for the T-set to be trivial modulo roots of unity.