Entropy and mixing for amenable group actions

For Gamma a countable amenable group consider those actions of Gamma as mea sure preserving transformations of a standard probability space, written as (T-gamma)(gamma is an element of Gamma) acting on (X, F, mu). We say (T-ga mma)(gamma is an element of Gamma) has completely positive entropy (or simp ly cpe for short) if for any finite and nontrivial partition P of X the ent ropy h(T, P) is not zero. Our goal is to demonstrate what. is well known fo r actions of Z and even Z(d), that actions of completely positive entropy h ave very strong mixing properties. Let S-i be a list of finite subsets of G amma. We say the S-i spread if any particular gamma not equal id belongs to at most finitely many of the sets SiSi-1. THEOREM 0.1. Ebr (T-gamma)(gamma is an element of Gamma) an action of Gamma of completely positive entropy and P any finite partition, for any sequenc e of finite sets S-i subset of or equal to Gamma which spread we have 1/#Si h (T-V(gamma is an element of Si)gamma-1(P))(i)-->h(P). The proof uses orbit equivalence theory in an essential way and represents the first significant application of these methods to classical entropy and mixing.