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Let  ϕ  :  M   n   →  B ( H ) be an injective completely positive contraction with ‖ ϕ  −1  :  ϕ ( M   n  ) →  M   n   ‖ cb  ⩽ 1 +  δ (ϵ). We show that if either C*( ϕ ( M   n  )) contains no non‐trivial compact operators or  ϕ ( M   n  ) approximately contains a rank 1 projection, then there is a complete order embedding  ψ  :  M   n   →  B ( H ) with ‖  ϕ  −  ψ  ‖  cb  < ϵ. 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  ℒ  ∞ ( 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.

Perturbations of completely positive maps and strong NF algebras