Monotone complete C*-algebras and generic dynamics

Kazuyuki Saito, J D Maitland Wright

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 languageEnglish
Pages (from-to)549-589
Number of pages41
JournalProceedings of the London Mathematical Society
Volume107
Issue number3
Early online date14 Feb 2013
DOIs
Publication statusPublished - Sep 2013

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C*-algebra
Monotone
Orbit
Stone Space
Orbit Equivalence
Monotonic increasing sequence
Cross product
Infinite Groups
Von Neumann Algebra
Boolean algebra
Equivalence relation
Envelope
Fermions
Subalgebra
Cardinality
Quotient
Semigroup
Isomorphic
Projection
Algebra

Cite this

Monotone complete C*-algebras and generic dynamics. / Saito, Kazuyuki; Wright, J D Maitland.

In: Proceedings of the London Mathematical Society, Vol. 107, No. 3, 09.2013, p. 549-589.

Research output: Contribution to journalArticle

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