On a tensor-analogue of the Schur product

K. Sumesh; V. S. Sunder

doi:10.1007/s11117-015-0377-x

paper

On a tensor-analogue of the Schur product

arXiv:1509.04884 · doi:10.1007/s11117-015-0377-x

Abstract

We consider the tensorial Schur product $R \circ^{\otimes} S = [r_{ij} \otimes s_{ij}]$ for $R \in M_{n} (A), S \in M_{n} (B),$ with $A, B$ 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 $ϕ : M_{n} \to M_{d}$ is completely positive if and only if $[ϕ (E_{ij})] \in M_{n} (M_{d})^{+}$ , where of course ${E_{ij} : 1 \leq i, j \leq n}$ denotes the usual system of matrix units in $M_{n} (:= M_{n} (C))$ . We also discuss some other corollaries of the main result.

Corrected typos, The final publication (with marginal changes) is available at Springer via http://dx.doi.org/[10.1007/s11117-015-0377-x]

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