Advances in Operator Theory

Positive map as difference of two completely positive or super-positive maps

Tsuyoshi Ando

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For a linear map from ${\mathbb M}_m$ to ${\mathbb M}_n$, besides the usual positivity, there are two stronger notions, complete positivity and super positivity. Given a positive linear map $\varphi$ we study a decomposition $\varphi = \varphi^{(1)} - \varphi^{(2)}$ with completely positive linear maps $\varphi^{(j)} (j = 1,2)$. Here $\varphi^{(1)} + \varphi^{(2)}$ is of simple form with norm small as possible. The same problem is discussed with super-positivity in place of complete positivity.

Article information

Adv. Oper. Theory, Volume 3, Number 1 (2018), 53-60.

Received: 25 February 2017
Accepted: 11 March 2017
First available in Project Euclid: 5 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47C15: Operators in $C^*$- or von Neumann algebras
Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 15A69: Multilinear algebra, tensor products

positive map completely positive map super-positive map norm tensor product


Ando, Tsuyoshi. Positive map as difference of two completely positive or super-positive maps. Adv. Oper. Theory 3 (2018), no. 1, 53--60. doi:10.22034/aot.1702-1129.

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