A factorization property of positive maps on C^*-algebras

The purpose of this short note is to clarify and present a general version of an interesting observation by Piani and Mora (Physic. Rev. A 75, 012305 (2007)), linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let A_i, C_i be unital C*-algebras and let α_i be positive linear maps from A_i to C_i, i=1,2. We obtain conditions under which any positive map β from the minimal C*-tensor product A₁ otimes_min A₂ to C₁ otimes_min C₂, such that α₁ otimes α₂ geq β, factorizes as β= γotimes α₂ for some positive map γ. In particular we show that when α_i colon A_i → B\(mathcal H_i\) are completely positive (CP) maps for some Hilbert spaces mathcal H_i \(i=1,2\), and α₂ is a pure CP map and β is a CP map so that α₁ otimes α₂ - β is also CP, then β= γotimes α₂ for some CP map γ. We show that a similar result holds in the context of positive linear maps when A₂ = C₂ = B\(mathcal H\) and α₂ = id. As an application we extend \cite[IX Theorem]{PM}revisited recently by Huber et al in cite{HLLM} to show that for any linear map τ from a unital C*-algebra A to a C*-algebra C, if τotimes id_k is decomposable for some k geq 2, where id_k is the identity map on the algebra M_k\(mathbb {C}\) of ktimes k matrices, then τ is completely positive.