  The generalized state space $ S_{\mathcal{H}}(\mathcal{\mathcal{A}})$ of all
unital completely positive (UCP) maps on a unital $C^*$-algebra $\mathcal{A}$
taking values in the algebra $\mathcal{B}(\mathcal{H})$ of all bounded
operators on a Hilbert space $\mathcal{H}$, is a $C^\ast$-convex set. In this
paper, we establish a connection between $C^\ast$-extreme points of
$S_{\mathcal{H}}(\mathcal{A})$ and a factorization property of certain algebras
associated to the UCP map. In particular, this factorization property of some
nest algebras is used to give a complete characterization of those
$C^\ast$-extreme maps which are direct sums of pure UCP maps. This
significantly extends a result of Farenick and Zhou [Proc. Amer. Math. Soc. 126
(1998)] from finite to infinite dimensional Hilbert spaces. Also it is shown
that normal $C^\ast$-extreme maps on type $I$ factors are direct sums of normal
pure UCP maps if and only if an associated algebra is reflexive. Further, a
Krein-Milman type theorem is established for $C^\ast$-convexity of the set $
S_{\mathcal{H}}(\mathcal{A})$ equipped with bounded weak topology, whenever
$\mathcal{A}$ is a separable $C^\ast$-algebra or it is a type $I$ factor. As an
application, we provide a new proof of a classical factorization result on
operator valued Hardy algebras.
