Extremal Unital Completely Positive Maps and Their Symmetries

摘要

We consider the set P-1(A, M) (respectively CP1(A, M) of unital positive (completely) maps from a C * algebra A to a von-Neumann sub-algebraMof B(H), the algebra of bounded linear operators on a Hilbert space H. We study the extreme points of the convex set P-1(A, M) (CP1(A, M)) via their canonical lifting to the convex set of (unital) positive (completely) normal maps from <^> A toM, where A** is the universal enveloping von-Neumann algebra overA. IfA = Mthen a (completely) positive map t admits a unique decomposition into a sum of a normal and a singular (completely) positive maps. Furthermore, if M is a factor then a unital complete positive map is a unique convex combination of unital normal and singular completely positive maps. We also used a duality argument to find a criteria for an element in the convex set of unital completely positive maps with a given faithful normal invariant state on M to be extremal. In our investigation, gauge symmetry in the minimal Stinespring representation of a completely positive map and Kadison theorem on order isomorphism played an important role.