Weakly strongly regular uniform algebras

TL;DR

The paper constructs specific uniform algebras on compact sets demonstrating nuanced ideal containment properties, confirming a conjecture and introducing a novel 'square root of Swiss cheese' construction.

Contribution

It proves a conjecture by Alexander Izzo by constructing uniform algebras with precise ideal containment behaviors using a new 'square root of Swiss cheese' method.

Findings

Existence of compact sets with prescribed ideal containment properties.

Confirmation of Izzo's conjecture on uniform algebra ideals.

Introduction of a new construction technique for uniform algebras.

Abstract

Given a uniform algebra A on a compact Hausdorff space X and a point x in X, denote by M_x the ideal of functions in A that vanish at x and by J_x the ideal of functions in A that vanish on a neighborhood of x. It is shown that for each integer m greater than or equal to 2, there exists a compact plane set K containing the origin such that in R(K) the closure of J_x contains M_x for every x in K minus {0} and the closure of J_0 contains M_0^m but does not contain M_0^{m-1}. This result establishes a recent conjecture of Alexander Izzo. For the proof we introduce a construction that could be described as taking square roots of Swiss cheeses.