Here is a theorem: Let $A$ be a separable unital $C^*$-algebra so that $A\subset B(l^2)$ with $1_A=1_\infty$, and let $\phi:A\to M_n(\mathbb C)\subset M_n(B(l^2))$ be a contractive completely positive linear map which vanishes on $A\cap \mathcal K$. Then there exists a sequence of contractions $V_k\in M_n(B(l^2))(\simeq B(l^2))$ such that $\|\phi(a)-V_k^*aV_k\|\to 0$ as $k\to \infty$ for all $a\in A$.
This is left as an excersice on the book I am reading. I have shown that there exists $V_k$ such that $\|\phi(a)-V_k^*\text{diag}(a,...,a)(=a\otimes 1_n)V_k\|\to 0$. But how do I show for $a\notin \mathcal K$ there is some $W$ such that $W^*aW-\text{diag}(a,...,a)\approx 0$?