Extremal unital completely positive normal maps and its symmetries

TL;DR

This paper investigates the structure and extremal properties of unital completely positive maps between $C^*$-algebras and von Neumann algebras, revealing their decomposition into normal and singular components using duality and symmetry techniques.

Contribution

It introduces a canonical lifting approach to analyze extremal points of unital positive maps and characterizes their decomposition into normal and singular parts.

Findings

Unique decomposition of positive maps into normal and singular components.

Criteria for extremal elements in the convex set of unital completely positive maps.

Role of gauge symmetry and Kadison theorem in the analysis.

Abstract

We consider the convex set of ( unital ) positive ( completely ) maps from a $C^*$ algebra $\cla$ to a von-Neumann sub-algebra $\clm$ of $\clb(\clh)$, the algebra of bounded linear operators on a Hilbert space $\clh$ and study its extreme points via its canonical lifting to the convex set of ( unital ) positive ( complete ) normal maps from $\hat{\cla}$ to $\clm$, where $\hat{\cla}$ is the universal enveloping von-Neumann algebra over $\cla$. If $\cla=\clm$ and a ( complete ) positive operator $\tau$ is a unique sum of a normal and a singular ( complete ) positive maps. Furthermore, a unital complete positive map is a unique convex combination of unital normal and singular complete positive maps. We used a duality argument to find a criteria for extremal elements in the convex set of unital completely positive maps having a given faithful normal invariant state. In our investigation, gauge symmetry in Stinespring representation and Kadison theorem on order isomorphism played an important role.