Monotone Complete C*-algebras and Generic Dynamics

TL;DR

This paper establishes a connection between group actions on Stone spaces and the construction of a large family of distinct, small, monotone complete Type III C*-algebras with specific dynamical properties.

Contribution

It introduces new dynamical results linking group actions to orbit equivalence and constructs a diverse family of non-isomorphic monotone complete C*-algebras with unique classification features.

Findings

Orbit equivalence can be realized via the Dyadic Group on a generic subset.

Constructed a large family of non-isomorphic, hyperfinite, Type III monotone complete C*-algebras.

Each algebra in the family is small, monotone complete, and not a von Neumann algebra.

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.