Perturbations of completely positive maps and strong NF algebras

Let $\phi:M_n\to B(H)$ be an injective, completely positive contraction with $\V\phi^{-1}:\phi(M_n)\to M_n\V_{cb}\leq1+\delta(\epsilon).$ We show that if either (i) $\phi(M_n)$ is faithful modulo the compact operators or (ii) $\phi(M_n)$ approximately contains a rank 1 projection, then there is a complete order embedding $\psi:M_n\to B(H)$ with $\V\phi-\psi\V_{cb}<\epsilon.$ We also give examples showing that such a perturbation does not exist in general. As an application, we show that every $C^*$-algebra $A$ with $\mathcal{OL}_\infty(A)=1$ and a finite separating family of primitive ideals is a strong NF algebra, providing a partial answer to a question of Junge, Ozawa and Ruan.