Monotone complete C*‐algebras and generic dynamics

By Kazuyuki Saito, J. Wright. 2012

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/26b25fdfc0885576145530dfac7b0fe2313552ce