Quantum Markov chains and fields

This book aims is to investigate quantum Markov chains (QMC) on Cayley trees. It proves the uniqueness of backward and forward QMC associated to the XY-model with competing interaction Ising model on the Cayley tree of order two. Proceeding from this fact, the following question naturally arises: how do these backward and forward QMC relate to each other? Thus, an observation of a relation between those QMC associated with the previous model was needed. We point out that this kind of model does not have one-dimensional analogous, i.e. the considered model persists only on trees. The uniqueness of backward and forward QMC associated to the XY-model with competing interaction Ising model on Cayley trees of order two means that the limit state does exist and does not depend on the boundary conditions, i.e. it is unique. We establish that the unique QMC has a clustering property, i.e. it is mixing with respect to translations of the tree. This implies that the von Neumann algebra πφ associated with GNS-representation of φ is a factor.