Monotone complete C*-algebras and generic dynamics

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 by an action of the Dyadic Group, ⊕ℤ2. As an application, we show that if M is the monotone cross-product C* -algebra, arising from the natural action of G on C(S), and if the projection lattice in C(S) is countably generated, then M can be approximated by an increasing sequence of finite-dimensional subalgebras. On each S, in a class considered earlier, we construct a natural action of ⊕ℤ2 with a free dense orbit. Using this we exhibit a huge family of small monotone complete C*-algebras, (Bλ, λ∈Λ) with the following properties. Each Bλ is a Type III factor that is not a von Neumann algebra. Each Bλ is a quotient of the Pedersen–Borel envelope of the Fermion algebra and hence is strongly hyperfinite. The cardinality of Λ is 2c, where c = 2ℵ0. When λ≠μ, then Bλ and Bμ take different values in the classification semi-group; in particular, they cannot be isomorphic.