Abstract
Let Ω⊂ℂn be a bounded domain and let \(\mathcal{A} \subset\mathcal{C}(\overline{\Omega})\) 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 \(\mathcal{A} = \mathcal {C}(\overline{\Omega})\) is that there is a holomorphic disk \(D \subset\overline{\Omega}\) such that all functions in F are holomorphic on D, i.e., the obvious obstruction is the only 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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Samuelsson, H., Wold, E.F. Uniform algebras and approximation on manifolds. Invent. math. 188, 505–523 (2012). https://doi.org/10.1007/s00222-011-0351-6
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DOI: https://doi.org/10.1007/s00222-011-0351-6