Abstract
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\( (\bar u) + \varepsilon \).
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This work was supported by National Natural Science Foundation of China (Grant No. 10771161)
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Fang, X., Zhao, Y. Approximately isometric lifting in quasidiagonal extensions. Sci. China Ser. A-Math. 52, 457–467 (2009). https://doi.org/10.1007/s11425-008-0117-9
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DOI: https://doi.org/10.1007/s11425-008-0117-9