CHAPTER 1 INTRODUCTION In the papers [14, 15] J. Feldman and C. C. Moore have introduced and studied the construction which can be named the "crossed product" of an abelian von Neumann algebra by an equivalence relation. This construction is analogous to the crossed product of a von Neumann algebra M by a group (see the example in Subsection 20.1 below). Furthermore, J. Feldman and C. C. Moore have developed the 2-cohomology theory for equivalence relations. This theory allowed them to prove the structure theorem on "crossed products" (see [15, Theorem 1]), i. e. to give the necessary and sufficient condition for a von Neumann algebra to be a "crossed product" of some abelian subalgebra by some equivalence relation. J. Feldman and C. C. Moore have also studied the automorphisms and isomor- phisms of such algebras. The equivalence relation in the construction of Feldman and Moore provides the algebra with coordinates i. e. allows the elements of the algebra to be represented as "generalized complex matrices". These coordinates were used by Feldman and Moore to study the automorphisms of these algebras. Subalgebras (in general non selfadjoint) of these algebras were studied by P. Muhly, K.-S. Saito and B. Solel [25]. Such subalgebras turned out to be as- sociated with partial orders. A key result in the study of these subalgebras is the Spectral theorem on bimodules (see [25, Theorem 2.5]) which allows one to associate bimodules over the diagonal M with subsets of the equivalence relation. Using coordinates for the study of operator algebras has proved very useful in several classes of algebras: e. g. CLS algebras (see [3]) and subalgebras of groupoid C*-algebras (see [27] in particular, for triangular AF algebras see [31]). In the paper [26] the authors proved a theorem on dilations of a representation of a subalgebra of such algebra to a representation of the whole algebra (see [26, Theorem 1]). In the present paper we introduce and study the construction which is a gen- eralization of the construction of J. Feldman and C. C. Moore to the case where the von Neumann algebra M is not commutative, but it is decomposed into the direct integral M Jx M(x)dfi(x). This integral may be the central decomposition (where all M(x) are factors), but it is not necessary. For this case we define and study the crossed product of a von Neumann algebra by an equivalence relation. We define also the analogue of the 2-cohomology theory in our case. This allows us to prove the structure theorem (see Theorem 12.2 below) which is the exact analogue of the structure theorem of J. Feldman and C. C. Moore. Received by the editor October 22, 1994. 1
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