Generic dynamics and monotone complete 𝐶*-algebras

Sullivan, Dennis; Weiss, B.; Wright, J. D. Maitland

doi:10.1090/S0002-9947-1986-0833710-X

Generic dynamics and monotone complete $C^ \ast$-algebras
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by Dennis Sullivan, B. Weiss and J. D. Maitland Wright PDF

Trans. Amer. Math. Soc. 295 (1986), 795-809 Request permission

Abstract:

Let $R$ be any ergodic, countable generic equivalence relation on a perfect Polish space $X$. It follows from the main theorem of $\S 1$ that, modulo a meagre subset of $X,R$ may be identified with the relation of orbit equivalence ensuing from a canonical action of ${\mathbf {Z}}$. Answering a longstanding problem of Kaplansky, Takenouchi and Dyer independently gave cross-product constructions of Type III $A{W^\ast }$-factors which were not von Neumann algebras. As a specialization of a much more general result, obtained in $\S 3$, we show that the Dyer factor is isomorphic to the Takenouchi factor.

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