Abstract
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 [Formula: see text] 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.
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