On a tensor-analogue of the Schur product

TL;DR

This paper introduces a tensorial analogue of the Schur product for matrices over $C^*$-algebras, proves its positivity preservation, and applies it to generalize Choi's theorem on complete positivity of linear maps.

Contribution

It defines a tensorial Schur product for matrices over $C^*$-algebras and uses it to extend classical results on complete positivity.

Findings

Tensorial Schur product of positive operators remains positive.

Generalization of Choi's theorem on complete positivity.

Discussion of additional corollaries of the main result.

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 $\mathcal{A}, \mathcal{B}$ 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 $\phi:M_n \to M_d$ is completely positive if and only if $[\phi(E_{ij})] \in M_n(M_d)^+$, where of course $\{E_{ij}:1 \leq i,j \leq n\}$ denotes the usual system of matrix units in $M_n (:= M_n(\mathbb{C}))$. We also discuss some other corollaries of the main result.