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arXiv · 1509.04884

On a tensor-analogue of the Schur product

K. Sumesh·V. S. Sunder
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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.

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K. Sumesh, V. S. Sunder. 2015-09-16. On a tensor-analogue of the Schur product. https://doi.org/10.1007/s11117-015-0377-x

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