Ergodic Equivalence Relations, Cohomology, and Von Neumann Algebras. II

Abstract

Let R be a Borel equivalence relation with countable equivalence classes, on the standard Borel space $(X,\mathcal {A},\mu )$. Let $\sigma$ be a 2-cohomology class on R with values in the torus $\mathbb {T}$. We construct a factor von Neumann algebra ${\mathbf {M}}(R,\sigma )$, generalizing the group-measure space construction of Murray and von Neumann [1] and previous generalizations by W. Krieger [1] and G. Zeller-Meier [1]. Very roughly, ${\mathbf {M}}(R,\sigma )$ is a sort of twisted matrix algebra whose elements are matrices $({a_{x,y}})$, where $(x,y) \in R$. The main result, Theorem 1, is the axiomatization of such factors; any factor M with a regular MASA subalgebra A, and possessing a conditional expectation from M onto A, and isomorphic to ${\mathbf {M}}(R,\sigma )$ in such a manner that A becomes the “diagonal matrices"; $(R,\sigma )$ is uniquely determined by M and A. A number of results are proved, linking invariants and automorphisms of the system (M, A) with those of $(R,\sigma )$. These generalize results of Singer [1] and of Connes [1]. Finally, several results are given which contain fragmentary information about what happens with a single M but two different subalgebras ${{\mathbf {A}}_1},{{\mathbf {A}}_2}$.