Outer Conjugacy for Actions of Continuous Amenable Groups

Abstract

The paper introduces a cohomological approach to the outer conjugacy problem in ergodic theory. Specifically, the following fact is proved: up to an isomorphism of an approximately finite type II ergodic full group there exists only one cohomological class of cocycles with dense range in a given amenable group. This result is used to establish outer conjugacy for strictly outer actions of continuous unimodular amenable subgroups of the normalizer of the full group generated by an ergodic type II automorphism. As a special case we show that outer conjugacy of compact groups reduces merely to conjugation. Besides that the correlation is established between the automorphisms of a principal groupoid with continuous orbits and ones of its discrete reduction. Also the correspondence is found between the automorphisms of groupoids with continuous orbits and those of the associated von Neumann algebras.