From the introduction: This large paper “treats the classification of measurable amenable ergodic actions of discrete groups, with emphasis on the case that the group be the integers. These are formulated in terms of matrix-valued random walks and their corresponding Poisson boundary. For a discrete group \(G\), we develop \(G\)-dimension spaces, a measure-theoretic version of dimension groups (given as direct limits approximating the preduals of \(W^*\) factors).
In particular, this leads naturally to an invariant which is a notion of rank, denoted \(\text{AT}(k)\), which extends approximate transitivity (which occurs exactly when \(k= 1\)). It also enables us to construct examples relatively easily-for example, answering a question of Thouvenot, an AT action of \(\mathbb{Z}\) with involution, for which the orbit space of the latter is ergodic but not AT (it is \(\text{AT}(2)\) however). We also construct an ergodic \(\text{AT}(2)\) but not AT action of the free group on two generators for which the corresponding \(G\)-matrix-valued random walk is temporally homogeneous (this is impossible for abelian groups, and probably for amenable groups).
Initial sections of the paper discuss the equivalence between the dimension space approach and the standard one. A transparent criterion for ergodicity is given in chapter three (3.9). Chapter four presents the ergodic \(\text{AT}(2)\) orbital action which is not AT that answers Thouvenot’s question. The technique is to show that telescoped products of a suitable family of \(2\times 2\) matrices (with entries polynomials with no negative coefficients) cannot be approximated by positive rank one matrices (with an appropriate notion of positive).
In contrast, in chapter five, numerical conditions are presented under which \(\text{AT}(2)\) actions are actually AT; these resemble those appearing in a well-known theorem of Mineka in a different context. Chapter six provides a lexicon for translating between classification of type III injective von Neuman factors and the classification in the paper. The following section deals with bounded type AT actions; using a theorem of LeCam, the classification boils down to that of sequences of compound Poisson distributions. The fixed point algebras under the natural involutions correspond to sequences of hyperbolic cosine distributions (in particular, they are AT), and using the results of section five, we can determine when the fixed point algebras are isomorphic to the original, and give criteria for the fixed point algebras to be themselves bounded. Chapter eight gives numerical criteria for the Poisson boundary of an AT action to be-trivial. Chapter nine deals with the construction mentioned-above of an ergodic \(\text{AT}(2)\) but not AT action of the free group on two generators.”