On the Approximation of Quasidiagonal C*-Algebras

Let A be a separable exact quasidiagonal C*-algebra. Suppose that π: A→L(H) is a faithful representation whose image does not contain nonzero compact operators. Then there exists a sequence ϕn: A→L(H) of completely positive contractions such that ‖π(a)−ϕn(a)‖→0 for all a∈A, and the C*-algebra generated by ϕn(A) is finite dimensional for each n. As an application it is shown that if the C*-algebra generated by a quasidiagonal operator T is exact and does not contain any nontrivial compact operator, then T is norm-limit of block-diagonal operators D=D1 ⊕D2⊕… with supi rank(Di)<∞.