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.