Recurrence, Transience, and Poisson Boundaries in Operator Algebras

The notions of recurrence and transience are fundamental tools in the study of the long term behavior of classical Markov chains; other long term properties are closely connected to them. However, these notions do not allow an immediate and unique generalization to quantum probability theory, and thus the study of their operator algebraic counterparts is still in its infancy.

This thesis contributes to this research by developing a coherent approach that incorporates existing concepts and methods. In particular, we refine and considerably extend the works of F. Haag and F. Fagnola, R. Rebolledo, and V. Umanità involving quantum extensions of (probabilistic) potential theory. The presented 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 quantum Markov operators and to a connection between recurrence, transience, and non-commutative Poisson boundaries.