Operator valued probability theory
In the first four sections an extended probability theory is developed. In the following two sections the corresponding category of extended probability spaces is introduced and basic categorical constructions corresponding to fundamental probability notions are described. An extended probability space consists of the Borel sets \(B(X)\) of a second countable locally compact Hausdorff topological space \(X\) and a suitable (sequentially continuous positive normed) operator valued measure \(F\) on \(B(X)\) into the real \(C^*\) algebra \(O(H)\) of bounded operators on the real Hilbert space \(H\). Besides other generalizations of probability notions, the authors have developed a notion of probability density for operator valued measures as an element in a Hilbert module. In the last section, products, dependence and independence are shown to be related to the monoidal structure of the category of extended probability spaces.
scientific article; zbMATH DE number 2119181