A General Method for Constructing Essential Uniform Algebras

TL;DR

This paper introduces a versatile method for constructing essential uniform algebras with specific properties, demonstrating new examples that challenge existing assumptions and highlight the importance of smoothness conditions.

Contribution

It provides a general construction technique for essential uniform algebras, producing examples that serve as counterexamples and illustrate the necessity of smoothness hypotheses.

Findings

Constructed an essential, natural, regular uniform algebra on the closed unit disc.

Created counterexamples to the peak point conjecture on manifolds of dimension at least three.

Developed an essential uniform algebra on the unit sphere in C^3 containing the ball algebra and invariant under the 3-torus.

Abstract

A general method for constructing essential uniform algebras with prescribed properties is presented. Using the method, the following examples are constructed: an essential, natural, regular uniform algebra on the closed unit disc; an essential, natural counterexample to the peak point conjecture on each manifold of dimension at least three; and an essential, natural uniform algebra on the unit sphere in C^3 containing the ball algebra and invariant under the action of the 3-torus. These examples show that a smoothness hypothesis in some results of Anderson and Izzo cannot be omitted.