Integrable, ergodic actions of Abelian groups on von Neumann algebras.

D. de Schreye

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Alexandre Danilenko, Toshihiro Hamachi (2000)

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The orbit equivalence of type $I I I_{0}$ ergodic equivalence relations is considered. We show that it is equivalent to the outer conjugacy problem for the natural trace-scaling action of a countable dense ℝ-subgroup by automorphisms of the Radon-Nikodym skew product extensions of these relations. A similar result holds for the weak equivalence of arbitrary type $I I I_{0}$ cocycles with values in Abelian groups.