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 $\alpha_i$ be positive linear maps from $A_i$ to $C_i,$ $i=1,2$. We obtain conditions under which any positive map $\beta$ from the minimal C*-tensor product $A_1 \otimes_{min} A_2$ to $C_1 \otimes_{min} C_2$, such that $\alpha_1 \otimes \alpha_2 \geq \beta$, factorizes as $\beta = \gamma \otimes \alpha_2$ for some positive map $\gamma$. In particular we show that when $\alpha_i \colon A_i \rightarrow B(\mathcal H_i)$ are completely positive (CP) maps for some Hilbert spaces $\mathcal H_i$ $(i=1,2)$, and $\alpha_2$ is a pure CP map and $\beta$ is a CP map so that $\alpha_1 \otimes \alpha_2 - \beta$ is also CP, then $\beta = \gamma \otimes \alpha_2$ for some CP map $\gamma$. We show that a similar result holds in the context of positive linear maps when $A_2 = C_2 = B(\mathcal H)$ and $\alpha_2 = id$. 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 $\tau$ from a unital C*-algebra $A$ to a C*-algebra $C$, if $\tau \otimes id_k$ is decomposable for some $k \geq 2$, where $id_k$ is the identity map on the algebra $M_k(\mathbb {C} )$ of $k\times k$ matrices, then $\tau$ is completely positive.