Entropy and mixing for amenable group actions

For Γ a countable amenable group consider those actions of Γ as measurepreserving transformations of a standard probability space, written as {Tγ}γ∈Γ acting on (X,F , μ). We say {Tγ}γ∈Γ has completely positive entropy (or simply cpe for short) if for any finite and nontrivial partition P of X the entropy h(T, P ) is not zero. Our goal is to demonstrate what is well known for actions of Z and even Zd, that actions of completely positive entropy have very strong mixing properties. Let Si be a list of finite subsets of Γ. We say the Si spread if any particular γ 6= id belongs to at most finitely many of the sets SiS i .