On a tensor-analogue of the Schur product

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 the...

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Published in:arXiv.org 2015-10
Main Authors: Sumesh, K, Sunder, V S
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description 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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title On a tensor-analogue of the Schur product
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