# On the structure of the set of positive maps

## TL;DR

This paper provides a comprehensive description of the set of positive maps between $C^*$-algebras and bounded operators, clarifying the origin of non-decomposable maps using tensor product theories.

## Contribution

It offers a new framework based on tensor product theories to characterize positive maps and explains the emergence of non-decomposable maps.

## Key findings

- Full description of positive maps set
- Clarification of non-decomposable maps origin
- New tensor-based characterization method

## Abstract

The full description of the set of positive maps $T: \qA \to \cB(\cH)$ ($\qA$ a $C^*$-algebra) is given. The approach is based on the simple prescription for selecting various types of positive maps. This prescription stems from the Grothendieck theory of projective tensor products complemented by the theory of tensor connes. In particular, the origin of non-decomposable maps is clarified.