A sharper Swiss cheese

TL;DR

This paper constructs specific compact planar sets to demonstrate the existence of nontrivial, strongly regular uniform algebras with additional properties, resolving longstanding open questions in the field.

Contribution

It proves the existence of a compact planar set with a nontrivial, strongly regular uniform algebra that is not weakly amenable, answering questions from decades ago.

Findings

Existence of a compact planar set with a nontrivial, strongly regular uniform algebra.

Construction of a uniform algebra with bounded relative units that is not weakly amenable.

Establishment of bounds on functions used in uniform algebra constructions.

Abstract

It is shown that there exists a compact planar set K such that the uniform algebra R(K) is nontrivial and strongly regular. This settles an issue raised by Donald Wilken 55 years ago. It is shown that the set K can be chosen such that, in addition, R(K) is not weakly amenable. It is also shown that there exists a uniform algebra that has bounded relative units but is not weakly amenable. These results answer questions raised by Joel Feinstein and Matthew Heath 17 years ago. A key ingredient in our proofs is a bound we establish on the functions introduced by Thomas Koerner to simplify Robert McKissick's construction of a nontrivial normal uniform algebra.