Entropy in the non-commutalrive dynamical systems and the structure of algebras

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1. In the reduced free product of C^*-algebras, (A,φ) = (A_1, φ_1) * (A_2,φ_2) with respect to faithful states φ_1 and φ_2, A is purely infinite and simple if A_1 is a reduced crossed product B ×_<a,r> G for G an infinite group, if φ_1 is well behaved with respect, to this crossed product decomposition, if A_2 ≠ C and if φ is not a trace.2. It is shown that for two dynamical approximation entropies (one C^* and one W^*) the implementing inner automorphism in a crossed product A ×_α Z has the same entropy value as the automorphism α. Using the techniques, in the proof, an example of a highly ergodic non-asymptotically abelian automorphism with topological entropy zero is also given. More specifically, it is shown that the free shifts on the Cuntz algebra Ο_∞ and the reduced free group C^*-algebra C^*_γ(F∞) have topological entropy zero.3. We defined an entropic invariant for automorphisms on amenable groups and investigated basic properties. Related results were obtained by Broen-Germain independently at the same time.4. We showed that the free group factor L(F_m) 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 restricting them to subgroups with Kazhdan's property T (for an example SL(n, Z)), we have a continuous family of non co-cycle conjugate outer actions on L(F_m),m 【greater than or equal】 2.