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 ...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 $ .展开更多
基金supported by National Natural Science Foundation of China (Grant No. 10771161)
文摘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 $ .
