Quasidiagonal extensions and \(AF\) algebras (Q1392476)

The purpose of this paper is to present some new results concerning the quasidiagonal algebras, AF and MF algebras, and the extension theory. Theorem 4.5. Consider a short exact sequence of \(C^*\)-algebras \[ 0\to A\to X\to F\to 0.\tag{\(*\)} \] If \(A\) is MF algebra and \(F\) is MF algebra then \(X\) is MF algebra. Corollary 4.6. Consider the sequence \((*)\) of separable nuclear \(C^*\)-algebras. If \(A\) is quasidiagonal and \(F\) is AF algebra then \(X\) is quasidiagonal. Theorem 5.4. Suppose that \((*)\) is a short exact sequence of \(C^*\)-algebras and \(F\) is AF algebra. If each \(D_n\) is a unital \(C^*\)-algebra of real rank zero, and if \(\phi: A\to (\prod D_n)/(\sum D_n)\) is a \(*\)-homomorphism such that the induced map on \(K_1(A)\) sends \(\partial(K_1(F))\) to \(0\), then there exist a \(*\)-homomorphism \(\psi: (\prod D_n)/(\sum D_n)\) extending \(\phi\).