I am stuck in problem 3.9.5 in the book of Brown and Ozawa (C$^*$-algebras and Finite-dimensional Approximations). The problem is:
Let $A\subset B(H)$ be exact and let $E\subset A$ be a finite dimensional operator system. Let $\{v_n\}$ be an orthonormal basis (assuming $H$ is separable) and $P_n$ the orthogonal projection onto the span of $\{v_1,...,v_n\}$. Show that for all large $n$ there exist u.c.p maps $\psi_n\colon P_nB(H)P_n\rightarrow B(H)$ such that $\psi_n(P_nxP_n)\in A$ and $\|x-\psi_n(P_nxP_n)\|\leq \epsilon$ for all $x\in E$.
If $A$ is nuclear, show one can force the entire range of $\psi_n$ into $A$.
I can solve the first part of this problem. In fact, it is not hard since we can use the conclusion in this section. Firstly, define $\phi_n: A\rightarrow P_nB(H)P_n$ by $\phi_n(x)=P_nxP_n$. Then for all large $n$, $\phi_n$ is injective on $E$. By Proposition. 3.9.6 and Lemma 3.9.7, one can find u.c.p maps $\psi_n: \phi_n(E)\rightarrow A$ such that $\|\psi_n-\phi_n^{-1}|_{\phi_n(E)}\|\rightarrow 0$. Then use the Arveson's extension theorem to get last $\psi_n$.
My question is when $A$ is nuclear, why we can force the range is totally in $A$? Thanks for all helps!