Uniform algebras and approximation on manifolds

TL;DR

This paper investigates when uniform algebras generated by holomorphic and pluriharmonic functions on complex manifolds equal all continuous functions, showing the only obstruction is the existence of a holomorphic disk where all generators are holomorphic.

Contribution

It generalizes previous results by characterizing the obstructions to uniform algebra equality on complex manifolds, including a bidisk maximality theorem.

Findings

The only obstruction is the existence of a holomorphic disk where all generators are holomorphic.

Generalization of Izzo's work to broader settings.

Extension of Wermer's maximality theorem to the bidisk boundary.

Abstract

Let $\Omega \subset \mathbb{C}^n$ be a bounded domain and let $\mathcal{A} \subset \mathcal{C}(\bar{\Omega})$ be a uniform algebra generated by a set $F$ of holomorphic and pluriharmonic functions. Under natural assumptions on $\Omega$ and $F$ we show that the only obstruction to $\mathcal{A} = \mathcal{C}(\bar{\Omega})$ is that there is a holomorphic disk $D \subset \bar{\Omega}$ such that all functions in $F$ are holomorphic on $D$, i.e., the only obstruction is the obvious one. This generalizes work by A. Izzo. We also have a generalization of Wermer's maximality theorem to the (distinguished boundary of the) bidisk.