Analytic Free Semigroup Algebras and Hopf Algebras

TL;DR

This paper demonstrates that analytic free semigroup algebras and their preduals possess Hopf algebra structures, revealing deep connections between their representations and corepresentations, and identifying their spectra.

Contribution

It establishes that both the algebra and its predual are Hopf algebras and uncovers their interconnected representation theories, a novel insight even in classical contexts.

Findings

Both $S$ and $S_*$ are Hopf algebras.

There is a bijection between representations of $S_*$ and corepresentations of $S$.

The spectrum of $S_*$ is explicitly identified.

Abstract

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.