Perturbations of completely positive maps and strong NF algebras

Abstract

Let $\varphi :M_n\to B(H)$ be an injective, completely positive contraction with $\Vϕ^{-1}:ϕ(M_n)\to M_n\V_{cb}\leq1+δ(ε).$ 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 $\Vϕ-ψ\V_{cb}<ε.$ 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 ${\mathcal{O}\mathcal{L}}_{\infty }(A)=1$ 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.

arXiv comment26 pages

References in corpus (2)

• On Gromov-Hausdorff convergence for operator metric spaces
• Irreducible representations of inner quasidiagonal C*-algebras

Cited by in corpus (1)

• The Ramsey property for Operator spaces and noncommutative Choquet simplices