Entropy of Automorphisms of II1-Factors Arising from the Dynamical Systems Theory

Let a countable amenable group G act freely and ergodically on a Lebesgue space (X, μ), preserving the measure μ. If T∈Aut(X, μ) is an automorphism of the equivalence relation defined by G then T can be extended to an automorphism αT of the II1-factor M=L∞(X, μ)⋊G. We prove that if T commutes with the action of G then H(αT)=h(T), where H(αT) is the Connes–Størmer entropy of αT, and h(T) is the Kolmogorov–Sinai entropy of T. We also prove that for given s and t, 0⩽s⩽t⩽∞, there exists a T such that h(T)=s and H(αT)=t.