Approximately isometric lifting in quasidiagonal extensions

作     者：FANG XiaoChun ZHAO YiLe 

作者机构：Department of MathematicsTongji UniversityShanghai 200092China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2009年第52卷第3期

页      面：457-467页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by National Natural Science Foundation of China (Grant No. 10771161) 

主　　题：commutativity lifting quasidiagonal extension 46L05 

摘      要：Let 0 → I → A → A/I → 0 be a short exact sequence of C*-algebras with A unital. Suppose that the extension 0 → I → A → A/I → 0 is quasidiagonal, then it is shown that any positive element (projection, partial isometry, unitary element, respectively) in A/I has a lifting with the same form which commutes with some quasicentral approximate unit of I consisting of projections. Furthermore, it is shown that for any given positive number ε, two positive elements (projections, partial isometries, unitary elements, respectively) $ \bar a,\bar b $ in A/I, and a positive element (projection, partial isometry, unitary element, respectively) a which is a lifting of $ \bar a $ , there is a positive element (projection, partial isometry, unitary element, respectively) b in A which is a lifting of $ \bar b $ such that ∥a?b∥ $ \left\| {\bar a - \bar b} \right\| + \varepsilon $ . As an application, it is shown that for any positive numbers ε and $ \bar u $ in U(A/I) 0 , there exists u in U(A)0 which is a lifting of $ \bar u $ such that cel(u) cel $ (\bar u) + \varepsilon $ .