Singly generated homogeneous 𝐹-algebras

Abstract

With each point m m in the spectrum of a singly generated F F -algebra we associate an algebra A m {A_m} of germs of functions. It is shown that if A m {A_m} is isomorphic to the algebra of germs of analytic functions of a single complex variable, then the spectrum of A A contains an analytic disc about m m . The algebra A A is called homogeneous if the algebras A m {A_m} are all isomorphic. If A A is homogeneous and none of the algebras A m {A_m} have zero divisors, we show that A A is the direct sum of its radical and either an algebra of analytic functions or countably many copies of the complex numbers. If A A is a uniform algebra which is homogeneous, then it is shown that A A is either the algebra of analytic functions on an open subset of the complex numbers or the algebra of all continuous functions on its spectrum.