A separable Brown-Douglas-Fillmore theorem and weak stability

We give a separable Brown-Douglas-Fillmore theorem. Let $A$ be a separable amenable $C^*$-algebra which satisfies the approximate UCT, $B$ be a unital separable amenable purely infinite simple $C^*$-algebra and $h_1, \, h_2: A\to B$ be two monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily equivalent if and only if $[h_1]=[h_2]\,\,\,\,{\rm in}\,\,\, KL(A,B).$ We prove that, for any $\varepsilon>0$ and any finite subset $\mathcal{F}\subset A$, there exist $\delta>0$ and a finite subset $\mathcal{G}\subset A$ satisfying the following: for any amenable purely infinite simple $C^*$-algebra $B$ and for any contractive positive linear map $L: A\to B$ such that

for all $a\in \mathcal{G},$ there exists a homomorphism $h: A\to B$such that

provided, in addition, that $K_i(A)$ are finitely generated. We also show that every separable amenable simple $C^*$-algebra $A$ with finitely generated $K$-theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple $C^*$-algebras. As an application, related to perturbations in the rotation $C^*$-algebras studied by U. Haagerup and M. Rørdam, we show that for any irrational number $\theta$ and any $\varepsilon>0$ there is $\delta>0$ such that in any unital amenable purely infinite simple $C^*$-algebra $B$ if

for a pair of unitaries, then there exists a pair of unitaries $u_1$ and $v_1$ in $B$ such that