The structure of C*-extreme points in spaces of completely positive linear maps on C*-algebras

If A is a unital C*-algebra and if H is a complex Hilbert space, then the set SH(A) of all unital completely positive linear maps from A to the algebra B(H) of continuous linear operators on H is an operator-valued, or generalised, state space of A. The usual state space of A occurs with the one-dimensional Hilbert space C. The structure of the extreme points of generalised state spaces was determined several years ago by Arveson [Acta Math. 123(1969), 141-224]. Recently, Farenick and Morenz [Trans. Amer. Math. Sec. 349(1997), 1725-1748] studied generalised state spaces from the perspective of noncommutative convexity, and they obtained a number of results on the structure of C*-extreme points. This work is continued in the present paper, and the main result is a precise description of the structure of the C*-extreme points of the generalised state spaces of A for all finite-dimensional Hilbert spaces H.