Tracial rank of C^*-crossed products of AF algebras by discrete groups

离散群 AF 代数的 C^* 交叉积的追踪排序

基本信息

批准号：
14540217
负责人：
OSAKA Hiryuki
金额：
$ 1.86万
依托单位：
Ritsumeikan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540217/
关键词：
Operator algebras C^*-algebras stable rank C^*-crossed products Rokhlin property The classification Theorem for simple C^*-algebras AF C^*-algebras C*-環論 C*-接合積単純C*-環の分類定理 AF C*-環 C*-環実階数

项目摘要

1. We define the tracial Rokhlin property for an action on a simple unital C^*-algebra in the joint research with N.C.Phillips, and study its basic properties. This properly is weaker than the Rokhlin property by Kishimoto. When A is a simple unital C^*-algebra of tracial rank zero and α∈Aut(A) has the tracial Rokhlin property, then the crossed product algebra C^*-(Z, A, α) is simple, and has real rank zero, stable rank one, and the Fundamental Comparison Property in the sense of Blackadar. It is still opened if C^*(Z, A, α) has tracial rank zero.2. We define the cyclic Rokhlin property for an action on a unital C^*-algebra in the joint work with H Lin, which is stronger than the tracial Rokhlin property. We proved that when A is a simple unital C^*-algebra of tracial rank zero and α∈Aut(A) has the cyclic tracial Rokhlin property, then the algebra C^*(Z, A, α) is simple until C^*-algebra of tracial rank zero. Moreover if A is separable with the UCT, then A is a AH algebra. We also proved that if an approximately inner a∈Aut(A) has tracial Rokhlin property, then α has the cyclic tracial Rokhlin property.3. Let G be a finite group and α an action from G to Aut(UHF). We proved in the joint work with T. Teruya that the crossed product algebra C^*(G, UHF, α) has topological rank more than or equal to 2. It is still opened if C^*(G, A, α) has stable rank 1 for any simple AF algebra A and any action a from G to Aut(A).

1。我们在与N.C. phillips的联合研究中定义了对简单单元C^* - 代数的作用的Tracial Rokhlin属性，并研究了其基本特性。当a是一个简单的单元c^* - 奇异等级的代数零，并且α∈Aut（a）具有奇特的rokhlin属性，则交叉产物代数C^* - （z，a，α）很简单，具有真实的等级零，稳定的等级一号，并且具有BlackAdar的基本比较属性。如果C^*（z，a，α）的曲折等级为零，则仍将打开。2。我们定义了与Hin联合工作中的单位C^* - 代数的循环rokhlin属性，该属性比tracial rokhlin属性更强。我们规定，当a是一个简单的单元c^* - 奇异级的代数零，并且α∈Aut（a）具有环状曲裂rokhlin属性，则简单的代数c^*（z，a，α）很简单，直到c^* - c^* - tracial Cank Zero的代数。而且，如果A与UCT分开，则A是AH代数。我们还证明，如果大约内部a∈Aut（a）具有纹状体rokhlin属性，则α具有环状循环rokhlin属性3。令G为有限的组，而α从G到AUT（UHF）的动作。我们在与T. teruya的联合工作中证明了交叉产物代数C^*（G，g，uhf，α）具有拓扑等级超过或等于2。如果C^*（g，a，α）对于任何简单的AF AF ARGEBRA a，并且从G到AUT（A）的任何简单的AF代数A和任何简单的AF AF ALGEBRA A和任何简单的动作（A），则仍将打开。