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。这恰如其分地弱于岸本的Rokhlin财产。当A是迹秩为零的单C^*-代数，且α∈(A)具有迹Rokhlin性质时，则交叉积代数C^*-(Z，A，α)是单的，且具有实秩零、稳定秩一和Blackadar意义下的基本比较性质.如果C^*(Z，A，α)有迹秩零，则它仍然是开的。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C^*-代数，且α∈aut(A)具有循环迹Rokhlin性质时，则C^*(Z，A，α)是单的，直到C^*-迹秩为零的代数.此外，如果A是与UCT可分的，则A是AH代数。证明了如果近似内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是从G到Aut(超高频)的作用。在与T.Teruya的合作中，我们证明了交叉积代数C^*(G，uhf，α)具有大于或等于2的拓扑秩.如果C^*(G，A，α)对任何单AF代数A和从G到Aut(A)的任何作用a都有稳定的秩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).