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.
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Notes
For an explanation of terms like CP (\(=\)completely positive) and operator system, the reader may consult [3], for instance.
References
Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975)
Choi, M.D., Effros, E.G.: Injectivity and operator spaces. J. Funct. Anal. 24(2), 156–209 (1977)
Pisier, G.: Introduction to Operator Space Theory. LMS Lecture Note Series 294. Cambridge University Press, Cambridge (2003)
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Sumesh, K., Sunder, V.S. On a tensor-analogue of the Schur product. Positivity 20, 621–624 (2016). https://doi.org/10.1007/s11117-015-0377-x
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DOI: https://doi.org/10.1007/s11117-015-0377-x