Boundary representations from constrained interpolation

Abstract

In this paper, we study $C^{\ast }$-envelopes of finite-dimensional operator algebras arising from constrained interpolation problems on the unit disc. In particular, we consider interpolation problems for the algebra $H_{\text{node}}^{\infty }$ that consists of bounded analytic functions on the unit disk that satisfy $f(0)=f(\hat{I}»)$ for some $0\ne \hat{I}»\in \mathbb{D}$. We show that there exist choices of four interpolation nodes that exclude both $0$ and $\hat{I}»$, such that if $I$ is the ideal of functions that vanish at the interpolation nodes, then $C_e^{\ast }(H_{\text{node}}^{\infty }/I)$ is infinite-dimensional. This differs markedly from the behavior of the algebra corresponding to interpolation nodes that contain the constrained points studied in the literature. Additionally, we use the distance formula to provide a completely isometric embedding of $C_e^{\ast }(H_{\text{node}}^{\infty }/I)$ for any choice of $n$ interpolation nodes that do not contain the constrained points into $M_n(G_{nc}^2)$, where $G_{nc}^2$ is Brown's noncommutative Grassmannian.

second version. Fixed typos, notations, and structure. Improved some arguments