A Peak Point Theorem for Uniform Algebras on Real-Analytic Varieties

TL;DR

This paper proves a new peak-point theorem for uniform algebras generated by real-analytic functions on real-analytic varieties, extending previous results and addressing a longstanding conjecture in the field.

Contribution

The authors establish a peak-point theorem for uniform algebras generated by real-analytic functions on real-analytic varieties, generalizing earlier work and providing new insights.

Findings

Established a peak-point theorem for real-analytic uniform algebras

Generalized previous results to broader classes of varieties

Extended understanding of peak points in uniform algebras

Abstract

It was once conjectured that if $A$ is a uniform algebra on its maximal ideal space $X$, and if each point of $X$ is a peak point for $A$, then $A = C(X)$. This peak-point conjecture was disproved by Brian Cole in 1968. Here we establish a peak-point theorem for uniform algebras generated by real-analytic functions on real-analytic varieties, generalizing previous results of the authors and John Wermer.