Singly generated homogeneous $F$-algebras

Abstract:

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