The extensions of $C^*$-algebras with tracial topological rank no more than one

Abstract

Let $0\rightarrow I\rightarrow A\rightarrow A/I \rightarrow 0$ be a short exact sequence of $C^*$-algebras with $A$ unital. Suppose that $I$ has tracial topological rank no more than one and $A/I$ is TAI (in particular, if $A/I$ is simple and has tracial topological rank no more than one). It will be proved that $A$ has tracial topological rank no more than one if the extension is quasidiagonal, and $A$ has the property ($P_1$) if the extension is tracially quasidiagonal.