Uniform algebras and approximation on manifolds

Let $Ω\subset \mathbb{C}^n$ be a bounded domain and let $\mathcal{A} \subset \mathcal{C}(\barΩ)$ be a uniform algebra generated by a set $F$ of holomorphic and pluriharmonic functions. Under natural assumptions on $Ω$ and $F$ we show that the only obstruction to $\mathcal{A} = \mathcal{C}(\barΩ)$ is that there is a holomorphic disk $D \subset \barΩ$ 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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