Noncommutative domains, universal operator models, and operator algebras

Let B (H) be the algebra of all bounded linear operators on a Hilbert space H. The main goal of the paper is to find large classes of noncommutative domains in B (H) with prescribed universal operator models, acting on the full Fock space with n generators, and to study these domains and their universal models in connection with the Hardy algebras and the C^*-algebras they generate. While the class of these domains contains the regular noncommutative domains previously studied in the literature, the main focus of the present paper is on the non-regular domains. The multi-variable operator theory of these domains is developed throughout the paper.