Dilation and Functional Models for Pure $\mathbfΘ_n$-Contractions and the von Neumann Inequality on Distinguished Varieties in $\mathbfΘ_n$

In this paper, we introduce the notion of a distinguished variety in the domain $\mathbfΘ_n$. One of the main results of the paper is a determinantal representation for every distinguished variety in $\mathbfΘ_n$. We also show that the closure of every distinguished variety is polynomially convex. Furthermore, we obtain a dilation and a functional model for a class of pure $\mathbfΘ_n$-contractions. Finally, we show that for a $\mathbfΘ_n$-contraction $\mathbf{T}=(T_1,\dots,T_n)$ such that $T_n^*$ is a pure contraction, there exists an algebraic variety in $\mathbfΘ_n$ for which the von Neumann inequality holds on the intersection of the closure of the variety with the distinguished boundary of $\mathbfΘ_n$.