Noncommutative Probability on von Neumann Algebras

We generalize ordinary probability theory to those von Neumann algebras A, for which Dye's generalized version of the Radon‐Nikodym theorem holds. This includes the classical case in which A is an Abelian von Neumann algebra generated by an observable or complete set of commuting observables. Via Gleason's theorem, this also includes the case of ordinary quantum mechanics, in which A=B(H) is the von Neumann algebra of all bounded operators on a separable Hilbert space H. Particular consideration is given to the concepts of conditioning, sufficient statistics, coarse‐graining, and filtering.