Role of Operator Algebras Arising From Sub Product Systems

Exploring Techniques in Non-self-adjoint Operator Algebras and Operator Systems

Abstract

In this paper we bring together several techniques in the theory of non-self-adjoint operator algebras and operator systems. We begin with classification of non-self-adjoint and self-adjoint operator algebras constructed from C*-correspondence and more specifically, from certain generalized Markov chains. We then transitions to the study of noncommutative boundaries in the sense of Arveson, and their use in the construction of dilations for families of operators arising from directed graphs. Finally, we discuss connections between operator systems and matrix convex sets and use dilation theory to obtain scaled inclusion results for matrix convex sets. We begin with classification of non-self-adjoint operator algebras. In Chapter 3 we solve isomorphism problems for tensor algebras arising from weighted partial dynamical systems.