A Coherent Approach to Recurrence and Transience for Quantum Markov Operators

We present a coherent approach to recurrence and transience, starting from a version of the Riesz decomposition theorem for superharmonic elements. Our approach allows straightforward proofs of some known results, entails new theorems, and has applications to other aspects of completely positive operators: It leads to a classification of idempotent Markov operators, thereby identifying concretely the Choi-Effros product, which can be introduced on the range of these maps. We obtain an abstract Poisson integral and a representation theorem for idempotent entanglement breaking channels.