CHAPTER 2 PRELIMINARIES Let M be a von Neumann algebra acting on a Hilbert space H. Let M be decom- posed into a direct integral M = Jx M(x)dfj,(x) where (X, 6, //) is a standard Borel measure space. Let H = Jx H{x)dn(x) be the corresponding decomposition of H. Let M possess a cyclic and separating vector ^ o = fx (f)o(x)d/j,(x). Our goal is to develop the Feldman-Moore construction [14, 15] for M in place ofL°°(X,tf,M). Let R be an equivalence relation on X. We will write x ~ y for (x,y) G it!. Let R be Borel as a subset of X x X. Let A = {(#, x) \ x G X} then A c R. Let nr : R ^ X, 7Ti : R -^ X, 6 : R — R be defined as follows: 7rr(x,y) = y, ni(x,y)=x, Q(x,y) = (y,x) Let i? be countable in the following sense: every coset of R is countable. Let v be a measure on R defined by: i/(C)= / c a r d t C H T r - 1 ^ ) ) ^ ) Jx for C Borel as a subset of X x X. This definition is correct by [14, Theorem 2], and / f(x,y)dv(x,y) = \ V f(x,y)dfi(y) Let £ be the set of all Borel isomorphisms of X with graphs in R, Ep be the set of all partial Borel isomorphisms of X with graphs in R. Let f be the collection of all maps / : i? — • \JyeX H(v) s u c n t n a t : 1. for every (x,y) G #: f(x,y) G #(y) 2. for every r G E the vector field a: H-» f(rx,x) belongs to H. Let for (x,y) G #: H(x,y) = #(?/). Then ({#(#, 2/)}(Xjy)€#,r) is a ^-measurable field of Hilbert spaces in the sense of Dixmier [10]. By [10], the field ({H(x1y)}^x^y)eR,T) defines the direct integral of Hilbert spaces: H = / H(x,y)dv{x,y) JR NOTATION. Let a — {tt(x,y)}(x,y)eR be a collection of isomorphisms of von Neumann algebras such that: 3

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