A characterization of the algebra of holomorphic functions on simply connected domain (Q1822729)

Let \(\Omega\) \(\subset {\mathbb{C}}\) be a simply connected domain. The algebra H(\(\Omega)\) of holomorphic functions on \(\Omega\) is an F-algebra. It is known that H(\(\Omega)\) has no (nonzero) topological divisors of zero and is singly-generated. The author shows that these last two propeties of H(\(\Omega)\) completely characterize it among F-algebras. He also uses to give a known characterization of the algebra of entire functions as a singly-generated Liouville algebra without topological divisors of zero.