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, ⊕ ℤ 2 . 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 ⊕ ℤ 2 with a free dense orbit. Using this we exhibit a huge family of small monotone complete C *‐algebras, ( B λ , λ∈Λ) with the following properties. Each B λ is a Type III factor that is not a von Neumann algebra. Each B λ is a quotient of the Pedersen–Borel envelope of the Fermion algebra and hence is strongly hyperfinite. The cardinality of Λ is 2 c , where c = 2 ℵ 0. When λ≠ μ , then B λ and B μ take different values in the classification semi‐group; in particular, they cannot be isomorphic.