Extreme points of the convex set of stochastic maps on a C*-algebra

Let A be a unital C*-subalgebra of the C*-algebra B(H) of all bounded opera tors on a complex separable Hilbert space H. Let S(A) denote the convex set of all unital, linear, completely positive and normal maps of A into itsel f. Using Stinespring's theorem, we present a criterion for an element T is an element of S(A) to be extremal. When A = B(H), this criterion leads to a n explicit description of the set of all extreme points of S(PI). We also o btain a quantum probabilistic analogue of the classical Birkhoff's theorem( 2) that every bistochastic matrix can be expressed as a convex combination of permutation matrices.