Uniform algebras and approximation on manifolds

Journal article, 2012

Let Ω⊂Cn be a bounded domain and let A⊂C(Ω¯) 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 A=C(Ω¯) is that there is a holomorphic disk D⊂Ω¯ 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.