Hilbert Modules in Probability

Hilbert modules are an invaluable tool in the study of dynamical systems both classical and quantum. Starting from the description of the universal Markov process of a semigroup of transition kernels in terms of Hilbert modules, motivating in this way carefully also how the notion emerges naturally, we illustrate some of the consequences. Several of these consequences relate a Markov process (that is, a classical irreversible dynamical system) to questions about irreversible and reversible quantum dynamical systems.