Extreme Points of the Convex set of Stochastic Maps on A C^*-Algebra

Let be a unital C^*-subalgebra of the C^*-algebra ℬ(ℋ) of all bounded operators on a complex separable Hilbert space ℋ. Let denote the convex set of all unital, linear, completely positive and normal maps of into itself. Using Stinespring's theorem, we present a criterion for an element to be extremal. When , this criterion leads to an explicit description of the set of all extreme points of . We also obtain a quantum probabilistic analogue of the classical Birkhoff's theorem² that every bistochastic matrix can be expressed as a convex combination of permutation matrices.