Perturbations of completely positive maps and strong NF algebras

Let $\varphi :M_n\to B(H)$ be an injective, completely positive contraction with $\text{V}{\varphi }^{-1}:\varphi (M_n)\to M_n{\text{V}}_{cb}\le 1+\delta (ϵ)\mathrm{.}$ We show that if either (i) $\varphi (M_n)$ is faithful modulo the compact operators or (ii) $\varphi (M_n)$ approximately contains a rank 1 projection, then there is a complete order embedding $\psi :M_n\to B(H)$ with $\text{V}\varphi -\psi {\text{V}}_{cb}<ϵ\mathrm{.}$ We also give examples showing that such a perturbation does not exist in general. As an application, we show that every $C^{\ast }$-algebra $A$ with ${\mathcal{O}\mathcal{L}}_{\mathrm{\infty }}(A)=1$ 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.