Executive Summary : | In view of the Poisson boundary theory of random walks, the notion of non-commutative Poisson boundaries was introduced by Masaki Izumi as a tool to investigate problems in operator algebras, more precisely, those in compact quantum group actions and in subfactor theory. Given a unital completely positive normal map on a von Neumann algebra to itself (such a map is called a non-commutative Markov operator), one can equip the fixed point set with an abstract von Neumann algebra structure, which we call the non-commutative Poisson boundary. In general, for a given Markov operator on a von Neumann algebra, it is a hard problem to find a concrete realization of the associated Poisson boundary, even in the commutative case. In the non-commutative case also, not many concrete realizations are known. This proposal primarily focuses on study of the von Neumann algebra structure and obtaining concrete realization of the non-commutative Poisson boundaries associated to `nice' Markov operators on certain classes of von Neumann algebras. For instance, we aim to construct `nice' classes of Markov operators on the algebra of bounded linear operators on full Fock space, Bosonic as well as Fermionic Fock spaces, the algebra of adjointable maps on W*-correspondences (especially, on Fock modules over Cuntz-Kriger bimodules of finite directed graphs) and then study the associated Poisson boundaries. Another interesting but quite challenging goal of this project is to construct examples of Markov operators acting on crossed product of a commutative von Neumann algebra by a `suitable' group such that the associated Poisson boundary is realized as a crossed product of a commutative von Neumann algebra by that group. It is noteworthy that any solution to the above problem yields examples of concrete realizations of Poisson boundaries. Further, in this project, we also focus on classification (as a von Neumann algebra) of a certain class of Poisson boundaries. |