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Positivity

, Volume 20, Issue 3, pp 621–624 | Cite as

On a tensor-analogue of the Schur product

  • K. Sumesh
  • V. S. SunderEmail author
Article

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.

Keywords

Choi’s theorem \(C^*\)-algebras Completely positive maps 

Mathematics Subject Classification

46.025 47.030 

References

  1. 1.
    Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Choi, M.D., Effros, E.G.: Injectivity and operator spaces. J. Funct. Anal. 24(2), 156–209 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Pisier, G.: Introduction to Operator Space Theory. LMS Lecture Note Series 294. Cambridge University Press, Cambridge (2003)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesChennaiIndia

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