Extensions of quasidiagonal $C^*$-algebras and K-theory

Abstract

Let $0 \to I \to E \to B \to 0$ be a short exact sequence of $C^*$-algebras where $E$ is separable, $I$ is quasidiagonal (QD) and $B$ is nuclear, QD and satisfies the UCT. It is shown that if the boundary map $\partial : K_1(B) \to K_0(I)$ vanishes then $E$ must be QD also.

A Hahn-Banach type property for $K_0$ of QD $C^*$-algebras is also formulated. It is shown that every nuclear QD $C^*$-algebra has this $K_0$-Hahn-Banach property if and only if the boundary map $\partial : K_1(B) \to K_0(I)$ (from above) always completely determines when $E$ is QD in the nuclear case.