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 $S_{H}(A)$ 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 ${\mathbb{C}}$. 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. Soc. 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 $H$.