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中文摘要：
给出了有单位元的纯无限单的C^＊一代数A通过Κ的扩张代数E的K-理论的一种刻画．证明了K0（E）同构于E中所有具有无限余投影的无限投影的Murry—von Neumann等价类全体所成的交换群，它还同构于上述投影的同伦等价类或酉等价类全体所成的交换群．还证明了对扩张代数E中的任一满的正元a，存在元索x∈E，使得x^＊ax=1，其中Κ为可分无限维Hilbert空间上紧算子全体所成的C^＊-代数．
英文摘要：
This paper describes K-theory for C^＊-algebra E which is the extension of a purely infinite simple C^＊-algebras A by Κ, the C^＊-algebra of all compact oprators on separable infinite dimensional Hilbert space. It is proved that K0（E） is the group of all Murry-von Neumann equivalent classes of all infinite projections in E with infinite complement projections, it is also equal to the group of all homotopy equivalent classes or unitary equivalent classes of the above projections. The authors also prove that for any full positive element a∈E, there exists an element x∈E, such that x^＊ax = 1.
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