Monotone Complete C*-algebras and Generic Dynamics

Abstract

Let S be the Stone space of a complete, non-atomic Boolean algebra. Let G be a countably infinite group of homeomorphisms of S. Let the action of G on S have a free dense orbit. Then we prove that, on a generic subset of S, the orbit equivalence relation coming from this action can also be obtained as an action of the (abelian) Dyadic Group. For the special case where the complete Boolean algebra is the algebra of regular open subsets of the real numbers, this reduces to a theorem of Sullivan-Weiss-Wright. By applying our new dynamical results we improve on an earlier paper by constructing a family of monotone complete C*-algebras, {B(t):t in T} with the following properties. First,T is large; it can be identified with the set of all subsets of the reals. Secondly each B(t) is a small C*-algebra,which is a monotone complete factor of Type III, is hyperfinite and is not a von Neumann algebra.Thirdly, when t is not equal to s, then B(s) is not isomorphic to B(t). In fact, B(s) and B(t) take different values in the classification semi-group for small monotone complete C*-algebras.