Localization for Uniform Algebras Generated by Real-Analytic Functions

TL;DR

This paper proves that under certain conditions, a uniform algebra generated by real-analytic functions on a compact set coincides with all continuous functions, resolving a specific case of a longstanding question in function algebra theory.

Contribution

It establishes that such a uniform algebra equals the continuous functions on the set, providing an affirmative answer to a 1965 question in the field.

Findings

The algebra generated by real-analytic functions equals C(K) under specified conditions.

The maximal ideal space of the algebra is the compact set K.

Every continuous function on K is locally approximable by algebra functions.

Abstract

It is shown that if $A$ is a uniform algebra generated by real-analytic functions on a suitable compact subset $K$ of a real-analytic variety such that the maximal ideal space of $A$ is $K$, and every continuous function on $K$ is locally a uniform limit of functions in $A$, then $A=C(K)$. This gives an affirmative answer to a special case of a question from the Proceedings of the Symposium on Function Algebras held at Tulane University in 1965.