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

1.在C^*-代数的减少自由产品中， （A，Σ） =（A_1， φ_1） * （A_2，φ_2）尊重忠实的状态φ_1和φ_2，如果A是纯无限的，简单地说，如果A_1是一个缩小的交叉产品B ×_<a，r> G代表G无限组，那么φ_1很好地受到尊重，从而实现这个交叉产品分解，if A_2/C and if/_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 ×_\Z has the same entropy value as the automorphism\。使用技术，在证明中，一个高度ergodic，非对称的亚伯兰自动化的例子也被赋予了。更具体地说，它显示了在昆茨代数O_∞和减少的自由组C^*-代数C^*_γ（F∞）上的自由转移。我们定义了一个针对可调节组和受调查的基本属性的非典型变量。Broen-Germain的相关结果独立于同一时间获得。我们展示了自由群体因子L（F_m）对所有m = 2， 3，...， ∞有一个连续的非共轭外行为家族，并给出了一个对Connes-Stormer行为在每个行为中出现的估计。如果限制他们与Kazhdan's T属性的子组（例如SL（n， Z）），我们在L（F_m），m [大于或等于] 2上有一个连续的非共循环共轭外行为家族。

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.