Uniform algebras and approximation on manifolds

Let $\mathrm{\Omega }\subset {\mathbb{C}}^n$ be a bounded domain and let $$\mathcal{A}$ \subset $\mathcal{C}$(\bar{\Omega})$beauniformalgebrageneratedbyaset$F$ofholomorphicandpluriharmonicfunctions\mathrm{.}Undernaturalassumptionson$\Omega$and$F$weshowthattheonlyobstructionto$$\mathcal{A}$ = $\mathcal{C}$(\bar{\Omega})$isthatthereisaholomorphicdisk$D \subset \bar{\Omega}$suchthatallfunctionsin$F$areholomorphicon$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.