AI 中文总结
本文研究了双均匀代数的插值序列，给出了插值序列的充分条件，并探讨了其几何特性与拓扑关系。
AI 中文摘要
给定一个双均匀代数$A=X^*$，其最大理想空间为$M_A$，我们提供了第一个以$A$的Gleason距离为条件的充分条件，用于判断$M_A∩X$中的序列是否为$A$的插值序列。我们证明了$\mathbb{D}^N$中的序列如果均匀分离当且仅当它是$H^\infty(\mathbb{D}^N)$的插值序列，并且其范数序列满足Blaschke条件，然后利用这一特征来根据其Gleason距离对$H^\infty(\mathbb{D}^N)$的插值序列进行分类。我们还研究了$\mathscr{H}^\infty$，即有界Dirichlet级数代数的插值序列，获得了一个序列在$\mathbb{C}+$中为该空间插值序列的必要和充分条件，并将此类序列的几何特性与$H^\infty(\mathbb{C}+)$的插值序列的几何特性联系起来。最后，我们证明了在第二偶数代数$A$的Shilov边界中的序列是$A^{**}$的插值序列当且仅当该序列在$w^*$拓扑下是离散的。
英文摘要
Given a dual uniform algebra $A=X^*$ with maximal ideal space $M_A$, we provide the first sufficient condition in terms of the Gleason distance of $A$ for a sequence in $M_A\cap X$ to be interpolating for $A$. We prove that a sequence in $\mathbb{D}^N$ is uniformly separated if and only if it is interpolating for $H^\infty(\mathbb{D}^N)$ and its sequence of norms satisfies the Blaschke condition, and then use this characterization to classify the interpolating sequences for $H^\infty(\mathbb{D}^N)$ in terms of its Gleason distance. We also study interpolating sequences for $\mathscr{H}^\infty$, the algebra of bounded Dirichlet series, obtaining necessary and sufficient conditions for a sequence in $\mathbb{C}+$ to be interpolating for this space, and relating the geometry of such sequences to that of the interpolating sequences for $H^\infty(\mathbb{C}+)$. Finally, we show that a sequence in the Shilov boundary of the second dual of a uniform algebra $A$ is interpolating for $A^{**}$ if and only if it is discrete for the $w^*$-topology.
