C*-Extreme Points of Positive Operator Valued Measures and Unital Completely Positive Maps

We study the quantum (C *) 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 *-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 *- 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 *-algebra with countable spectrum (in particular Cn) is C *-extreme in the set of unital completely positive maps if and only if it is a unital *-homomorphism.