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 , A , μ ) (X,\mathcal {A},\mu ) . Let σ \sigma be a 2-cohomology class on R with values in the torus T \mathbb {T} . We construct a factor von Neumann algebra M ( R , σ ) {\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, M ( R , σ ) {\mathbf {M}}(R,\sigma ) is a sort of twisted matrix algebra whose elements are matrices ( a x , y ) ({a_{x,y}}) , where ( x , y ) ∈ R (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 M ( R , σ ) {\mathbf {M}}(R,\sigma ) in such a manner that A becomes the “diagonal matrices"; ( R , σ ) (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 , σ ) (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 A 1 , A 2 {{\mathbf {A}}_1},{{\mathbf {A}}_2} .