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

1. 在与N.C.Phillips的联合研究中，我们定义了简单单位C^*-代数上的一个作用的迹迹Rokhlin性质，并研究了它的基本性质。这比岸本的Rokhlin性质弱。当A是一个迹秩为0且α∈Aut(A)具有迹秩Rokhlin性质的简单一元C^*-代数时，则交叉积代数C^*-（Z， A, α）是简单的，且具有实数秩为0，稳定秩为1，并且具有Blackadar意义上的基本比较性质。如果C^*（Z， A, α）的跟踪秩为0 .2，则仍然打开。在与H Lin的联合工作中，我们定义了单位C^*-代数上的作用的循环Rokhlin性质，它比迹迹Rokhlin性质更强。证明了当A是一个简单的C^*-迹秩为0的一元代数，且α∈Aut(A)具有循环迹秩Rokhlin性质时，则代数C^*（Z， A, α）是简单的，直到C^*-迹秩为0的代数为止。如果A与UCT是可分离的，则A是一个AH代数。我们还证明了如果一个近似内的a∈Aut(a)具有迹迹Rokhlin性质，则α具有循环迹迹Rokhlin性质。设G为有限群，α为从G到Aut（UHF）的作用。我们在与T. Teruya的联合工作中证明了交叉积代数C^*（G， UHF, α）的拓扑秩大于等于2。如果C^*（G， A, α）对任何简单AF代数A和从G到Aut(A)的任何作用A具有稳定的秩1，则它仍然是开的。

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).