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

非交换动力系统中的熵和代数结构

基本信息

批准号：
12640209
负责人：
CHODA Marie
金额：
$ 1.47万
依托单位：
Osaka Kyoiku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640209/
关键词：
Operator algebras Automorphism Non-commutative dynamical system Entropy State C^*-algebra Crossed product Free product

项目摘要

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.

1.在C^*-代数的约化自由积中，（A，φ）=（A_1，φ_1）*（A_2，φ_2）关于忠实态φ_1和φ_2，A是纯无限单的，如果A_1是一个约化的交叉积B ×_<a，r> G，G是一个无限群，如果φ_1关于这个交叉积分解是良好的，如果A_2 <$C且φ不是迹.证明了对于两个动力学近似熵（一个C^* 和一个W^*），交叉积A × α Z中的实现内自同构与自同构α具有相同的熵值.利用这些技巧，给出了拓扑熵为零的高度遍历非渐近阿贝尔自同构的一个例子。更具体地说，证明了Cuntz代数O_∞和约化自由群C^*-代数C^*_γ（F_∞）上的自由移位的拓扑熵为零.定义了顺从群上自同构的熵不变量，并研究了其基本性质。Broen-Germain同时独立获得了相关结果.本文证明了自由群因子L（F_m）对m = 2，3，···，∞有GL（n，Z）的一个连续的非共轭外作用族，并给出了作用族中每个自同构的Connes-Stormer熵的估计.通过将它们限制到具有Kazhdan性质T的子群（例如SL（n，Z）），我们得到了L（F_m）上的非余圈共轭外作用的连续族，m [大于或等于] 2。