On the Relationships Between the Moments of a POVM and the Generator of the von Neumann Algebra It Generates

In the present paper, we review some recent results about commutative positive operator valued measures (POVMs) and single out some open problems. We introduce a conjecture about the extension of some recent results and prove some important consequences of such conjecture. In particular, we prove that it implies the universal character of some of the mathematical objects we introduce, i.e., the fact that they do not depend on the POV measure we are considering. We analyze the relevance of this result. Finally, we point out that some of the results we present admit a constructive proof and we show the relevance of this fact to the theory of commutative POV measures.