A characterization of the disk algebra
The authors show that a complex unital uniform algebra is isomorphic to the disk algebra \(A(\mathbb D)\) if and only if every closed unital subalgebra with one generator is isomorphic to the whole algebra. Moreover, every such subalgebra of \(A(\mathbb D)\) is isometrically isomorphic to \(A(\mathbb D)\). Along the way, they prove the following interesting result: Let \(f\in A(\mathbb D)\) be non-constant. Then every component of the complement of \(f(\overline {\mathbb D})\) is a Jordan domain. The proof uses the theory of prime ends from the theory of conformal mappings.
scientific article; zbMATH DE number 1839961