### 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 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.

Original language | English |
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Pages (from-to) | 549-589 |

Number of pages | 41 |

Journal | Proceedings of the London Mathematical Society |

Volume | 107 |

Issue number | 3 |

Early online date | 14 Feb 2013 |

DOIs | |

Publication status | Published - Sep 2013 |

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## Cite this

Saito, K., & Wright, J. D. M. (2013). Monotone complete C*-algebras and generic dynamics.

*Proceedings of the London Mathematical Society*,*107*(3), 549-589. https://doi.org/10.1112/plms/pds084