Extensions by simple $C^{\ast }$-algebras -- Quasidiagonal extensions

Let $A$ be an amenable separable \CA and $B$ be a non-unital but $\sigma$-unital simple \CA with continuous scale. We show that two essential extensions ${\tau }_1$ and ${\tau }_2$ of $A$ by $B$ are approximately unitarily equivalent if and only if $$[{\tau }_1]=[{\tau }_2]\mathrm{i}\mathrm{n}KL(A,M(B)\mathrm{/}B)\mathrm{.}$$ If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL(A,M(B)\mathrm{/}B)\mathrm{.}$ Using $KL(A,M(B)\mathrm{/}B),$ we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.