Uniform algebras and approximation on manifolds

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.

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