解析自由半群代数与霍普夫代数
Analytic Free Semigroup Algebras and Hopf Algebras
摘要：
设$\fS$是解析自由半群代数.本文研究了$\fS$及其前对偶$\fS_*$的更丰富的结构。证明了$\fS$和$\fS_*$都是霍普夫代数.此外，$\fS$和$\fS_*$的结构是紧密相连的：一方面，$\fS_*$的完全有界表示集和$\fS$的上表示集之间存在一个双射，另一方面，$\fS$可以从$\fS_*$的完全有界表示的系数算子中恢复出来.作为一个有趣的应用，我们的结果，（Gelfand）光谱的$\fS_*$确定。令人惊讶的是，本文的主要结果似乎新的，即使在经典的情况下。
英文摘要：
Let $\fS$ be an analytic free semigroup algebra. In this paper, we explore richer structures of $\fS$ and its predual $\fS_*$. We prove that $\fS$ and $\fS_*$ both are Hopf algebras. Moreover, the structures of $\fS$ and $\fS_*$ are closely connected with each other: There is a bijection between the set of completely bounded representations of $\fS_*$ and the set of corepresentations of $\fS$ on one hand, and $\fS$ can be recovered from the coefficient operators of completely bounded representations of $\fS_*$ on the other hand. As an amusing application of our results, the (Gelfand) spectrum of $\fS_*$ is identified. Surprisingly, the main results of this paper seem new even in the classical case.
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