Analytic Discs and Uniform Algebras on Real-Analytic Varieties

TL;DR

This paper proves that uniform algebras generated by real-analytic functions are either maximal or contain a disc where all functions are holomorphic, strengthening previous results in the field.

Contribution

It establishes a dichotomy for such algebras, showing they are either the entire continuous functions or have a holomorphic disc, under very general conditions.

Findings

Uniform algebras generated by real-analytic functions are either all continuous functions or contain a holomorphic disc.

The result generalizes and strengthens earlier theorems in the area.

Provides a clear criterion for the structure of these algebras.

Abstract

Under very general conditions it is shown that if $A$ is a uniform algebra generated by real-analytic functions, then either $A$ consists of all continuous functions or else there exists a disc on which every function in $A$ is holomorphic. This strengthens several earlier results concerning uniform algebras generated by real-analytic functions.