On a tensor-analogue of the Schur product

Abstract

We consider the tensorial Schur product $R \circ ^\otimes S = [r_{ij} \otimes s_{ij}]$ for $R \in M_n(\mathcal {A}), S\in M_n(\mathcal {B}),$ with $\mathcal {A}, \mathcal {B}~\text{ unital }~ C^*$-algebras, verify that such a ‘tensorial Schur product’ of positive operators is again positive, and then use this fact to prove (an apparently marginally more general version of) the classical result of Choi that a linear map $\phi :M_n \rightarrow M_d$ is completely positive if and only if $[\phi (E_{ij})] \in M_n(M_d)^+$, where of course $\{E_{ij}:1 \le i,j \le n\}$ denotes the usual system of matrix units in $M_n (:= M_n(\mathbb C))$. We also discuss some other corollaries of the main result.