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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.在与N.C.菲利普斯的合作研究中，我们定义了单单位C^*-代数上作用的迹Rokhlin性质，并研究了它的基本性质.这个性质比Kishimoto的Rokhlin性质弱。当A是迹秩为零的单单位C^*-代数且α∈Aut（A）具有迹Rokhlin性质时，则交叉积代数C^*-（Z，A，α）是单的，且具有真实的秩零、稳定秩一和Blackadar意义下的基本比较性质.如果C^*（Z，A，α）的迹秩为零，则它仍然是开的。在与H Lin的合作中，我们定义了单位C^*-代数上作用的循环Rokhlin性质，它比迹Rokhlin性质更强。证明了当A是迹秩为零的单单位C^*-代数且α∈Aut（A）具有循环迹Rokhlin性质时，代数C^*（Z，A，α）是单的直到迹秩为零的C^*-代数.此外，如果A可与UCT分离，则A是AH代数。证明了若a∈Aut（A）具有迹Rokhlin性质，则α具有循环迹Rokhlin性质.设G是有限群，α是G到Aut（UHF）的作用.我们在与T. Teruya证明了交叉积代数C^*（G，UHF，α）的拓扑秩大于或等于2.若对任意单AF代数A和任意从G到Aut（A）的作用a，C^*（G，A，α）具有稳定秩1，则该定理仍是开的.