$C^{\ast }$-extreme points of positive operator valued measures and unital completely positive maps

We study the quantum ($C^{\ast }$) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, $C^{\ast }$-extreme points of normalized POVMs on countable spaces (in particular for finite sets) are always spectral measures (normalized projection valued measures). More generally it is shown that atomic $C^{\ast }$-extreme points are spectral. A Krein-Milman type theorem for POVMs has also been proved. As an application it is shown that a map on any commutative unital $C^{\ast }$-algebra with countable spectrum (in particular ${\mathbb{C}}^n$) is $C^{\ast }$-extreme in the set of unital completely positive maps if and only if it is a unital $\ast$-homomorphism.