Full extensions and approximate unitary equivalences

Let $A$ be a unital separable amenable \CA and $C$ be a unital \CA with certain infinite property. We show that two full monomorphisms $h_1, h_2: A\to C$areapproximatelyunitarilyequivalentifandonlyif$[h_1]=[h_2]$in$KL(A,C).$Let$B$beanon-unitalbut$\sigma$-unital\text{CA}forwhich$M(B)/B$ has the certain infinite property. We prove that two full essential extensions are approximately unitarily equivalent if and only if they induce the same element in $KL(A,M(B)\mathrm{/}B)\mathrm{.}$ The set of approximately unitarily equivalence classes of full essential extensions forms a group. If $A$ satisfies the Universal Coefficient Theorem, it is can be identified with $KL(A,M(B)\mathrm{/}B)\mathrm{.}$