Entropy on non-commutative dynamical systems and the structure of operator algebras

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1)An entropical invariant is defined for automorphisms of countable discrete amenable groups, and it is shown relations between two entropies for an automorphism on the C^*-crossed product algebra and for its restriction to the original algebra.As an application to the free product^*, the topological entropy ht(θ) of an automorphism θ satishies that ht(θ^*θ^*......^*θ) = ht(θ^*1).2)We showed that the free group factor L(Fm) has a continuous family of non conjugate outer actions of GL(n, Z) for all m=2, 3,..., and give an estimation of the Connes-Stormer entropy for each automorphism appearing in the actions.By restrictng them to subgroups with Kazhdan's property T (for an example SL(n, Z), we have a continuous family of non cocycle conjugate outer actions on L(Fm), m>1.3)Let G be a discrete subgroup of the automorphism group of an operator algebra A. Assume that G is exact, that is, it admits an amenable action on some compact space.Then the entropy of an automorphism of the algebra A does not change by the canonical extension to the crossed product of A by G.This is shown for the topological entropy of an exact C^* algebra A and for the dynamical entropy of an AFD von Neumann algebra A.