  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.
