Extremal positive maps on $M_{3}(\mathbb{C})$ and idempotent matrices

TL;DR

This paper introduces a new method for analyzing extremal positive bistochastic maps on 3x3 complex matrices using convex set techniques and idempotent matrices, classifying extremal maps into two main categories.

Contribution

It develops a novel approach employing convex analysis and idempotent matrices to classify extremal positive maps on M_3, revealing two main categories based on simple idempotents.

Findings

Extremal positive bistochastic maps are classified into two categories.

All such maps not being Jordan isomorphisms are associated with specific idempotent matrices.

Norm conditions for extremal maps are established.

Abstract

A new method of analysing positive bistochastic maps on the algebra of complex matrices $M_{3}$ has been proposed. By identifying the set of such maps with a convex set of linear operators on $\mathbb{R}^{8}$, one can employ techniques from the theory of compact affine semigroups to obtain results concerning asymptotic properties of positive maps. It turns out that the idempotent elements play a crucial role in classifying the convex set into subsets, in which representations of extremal positive maps are to be found. It has been show that all positive bistochastic maps, extremal in the set of all positive maps of $M_{3}$, that are not Jordan isomorphisms of $M_3$ are represented by matrices that fall into two possible categories, determined by the simplest idempotent matrices: one by the zero matrix, and the other by a one dimensional orthogonal projection. Some norm conditions for matrices representing possible extremal maps have been specified and examples of maps from both categories have been brought up, based on the results published previously.