## Advances in Operator Theory

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

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

#### Abstract

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

**Source**

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

**Dates**

Received: 25 February 2017

Accepted: 11 March 2017

First available in Project Euclid: 5 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aot/1512497952

**Digital Object Identifier**

doi:10.22034/aot.1702-1129

**Mathematical Reviews number (MathSciNet)**

MR3730339

**Zentralblatt MATH identifier**

06804316

**Subjects**

Primary: 47C15: Operators in $C^*$- or von Neumann algebras

Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 15A69: Multilinear algebra, tensor products

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aot/1512497952