迹稳定秩1的C^＊-代数的稳定有限性与弱迹稳定秩1

Stable Finite Property of C^＊-Algebras with Tracial Stable Rank One Weak Tracial Stable Rank One

要给出了迹稳定秩1的C^＊-代数的稳定有限性,证明了如果A是有单位元迹稳定秩1的C^＊-代数,则A是稳定有限的,引入了弱迹稳定秩1的定义,并且证明了如果有单位元的C^＊-代数A是迹稳定秩1的,则A是弱迹稳定秩1的．对于单的具有SP性质的有单位元的C^＊-代数A,如果A是弱迹稳定秩1的,则A是迹稳定秩1的．同时给出了迹稳定秩1的C^＊-代数的一个等价条件,证明了一个有单位元的可分的C^＊-代数A是迹稳定秩1的,等价于A=（t4）limn→∞（An,pn）,其中tsr（An）=1．

This paper gives the stable finite property of C^＊-algebras with tracial stable rank one. It is showed that a unital C^＊-algebra A with tracial stable rank one is stable finite, and introduces the definition of weak tracial stable rank one, and proves that if a unital C^＊-algebra A with tracial stable rank one is weak tracial stable rank one, for simple unital C^＊-algebra A which has SP property with weak tracial stable rank one is tracial stable rank one. The authors also give a equivalent definition on C^＊-algebras with tracial stable rank one, it is showed that a separable unital C^＊-algebra A with tracial stable rank one if and only if A=（t4）limn→∞（An,pn）with tsr（An）=1.