Smooth and non-smooth $AF$-algebras and problem on invariant measures

Abstract

We separate the $AF$-algebras (correspondingly action of the countable groups on Cantor sets) onto two classes ---- "completely smooth" for which the set of all indecomposable traces (correspondingly list of all invariant ergodic measures) has nice parametrization, and the second class --- "completely non-smooth" for which the set of all traces (correspondingly set of all invariant measures) is Poulsen simplex and therefore there is no suitable parametrization of indecomposable traces, (ergodic measures) e.g.Choquet boundary. Important example of the first type of $AF$-algebra is group algebra of infinite symmetric group, and of the second type --- group algebra of some locally finite solvable group. The questions of recognition of those cases are closely related to the orbit theory of dynamical systems and theory of filtrations of sigma-fields in measure spaces and Borel spaces.