Kolmogorov's existence theorem for Markov processes in C^∗ algebras

Abstract

Given a family of transition probability functions between measure spaces and an initial distribution Kolmogorov's existence theorem associates a unique Markov process on the product space. Here a canonical non-commutative analogue of this result is established for families of completely positive maps between C^∗ algebras satisfying the Chapman-Kolmogorov equations. This could be the starting point for a theory of quantum Markov processes.