ENTROPY FOR CANONICAL SHIFTS AND STRONG AMENABILITY

When N⊂M is an inclusion of type II1 factors with finite index, let Γ be the canonical shift on the von Neumann algebra R generated by the derived tower of N⊂M, H(R|Γ(R)) the relative entropy, and H(Γ) the dynamical entropy. We characterize the equality cases of the inequalities in connection with the subexponential growth condition and the strong amenability condition. Also the topological entropy of Γ and the convergence of entropy increments on the derived tower are discussed.