标题： 约束插值的边界表示 显示英文标题

标题： Boundary representations from constrained interpolation

摘要： 在本文中，我们研究由单位圆上的约束插值问题产生的有限维算子代数的$C^*$包络。 特别是，我们考虑代数$H^\infty_{\text{node}}$的插值问题，该代数由满足$ f(0) = f(\lambda)$的单位圆盘上有界解析函数组成，其中对于某个$0 \neq \lambda \in \mathbb{D}$。 我们证明存在一些插值节点的选择，这些节点既不包含$0$也不包含$\lambda$，使得如果$I$是在插值节点处消失的函数的理想，那么$C^*_e(H^\infty_{\text{node}}/I)$是无限维的。 这与文献中研究的包含约束点的插值节点对应的代数行为有明显不同。 此外，我们使用距离公式为$C^*_e(H^\infty_{\text{node}}/I)$提供一个完全等距嵌入，对于任何不包含约束点的$n$插值节点的选择，将其嵌入到$M_n(G^2_{nc})$中，其中$G^2_{nc}$是布朗的非交换格拉斯曼流形。 显示英文摘要

摘要： 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.