Boundary representations from constrained interpolation

TL;DR

This paper investigates the $C^*$-envelopes of finite-dimensional operator algebras from constrained interpolation problems on the unit disk, revealing new infinite-dimensional behaviors and embeddings related to specific interpolation nodes.

Contribution

It demonstrates the existence of infinite-dimensional $C^*$-envelopes for certain constrained interpolation algebras and provides a completely isometric embedding into noncommutative Grassmannian spaces.

Findings

Existence of infinite-dimensional $C^*$-envelopes for specific interpolation node choices.

Distinct behavior when interpolation nodes exclude constrained points.

Embedding of $C^*$-envelopes into noncommutative Grassmannian spaces.

Abstract

In this paper, we study $C^*$-envelopes of finite-dimensional operator algebras arising from constrained interpolation problems on the unit disc. In particular, we consider interpolation problems for the algebra $H^\infty_{\text{node}}$ that consists of bounded analytic functions on the unit disk that satisfy $ f(0) = f(\lambda)$ for some $0 \neq \lambda \in \mathbb{D}$. We show that there exist choices of four interpolation nodes that exclude both $0$ and $\lambda$, such that if $I$ is the ideal of functions that vanish at the interpolation nodes, then $C^*_e(H^\infty_{\text{node}}/I)$ 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 $C^*_e(H^\infty_{\text{node}}/I)$ for any choice of $n$ interpolation nodes that do not contain the constrained points into $M_n(G^2_{nc})$, where $G^2_{nc}$ is Brown's noncommutative Grassmannian.