Extremal unital completely positive normal maps and its symmetries

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.