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 i...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 , two positive elements (projections, partial isometries, unitary elements, respectively) aˉ, ˉb in A/I, and a positive element (projection, partial isometry, unitary element, respectively) a which is a lifting of aˉ, there is a positive element (projection, partial isometry, unitary element, respectively) b in A which is a lifting of ˉb such that ab < aˉˉb + . As an application, it is shown that for any positive numbers and uˉ in U(A/I)0, there exists u in U(A)0 which is a lifting of uˉ such that cel(u) < cel(uˉ) + .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 , two positive elements (projections, partial isometries, unitary elements, respectively) aˉ, ˉb in A/I, and a positive element (projection, partial isometry, unitary element, respectively) a which is a lifting of aˉ, there is a positive element (projection, partial isometry, unitary element, respectively) b in A which is a lifting of ˉb such that ab < aˉˉb + . As an application, it is shown that for any positive numbers and uˉ in U(A/I)0, there exists u in U(A)0 which is a lifting of uˉ such that cel(u) < cel(uˉ) + .
