A sharper Swiss cheese

TL;DR

The paper resolves longstanding questions about strong regularity and weak amenability in uniform algebras by constructing a Swiss cheese set $K$ in the plane for which $R(K)$ is nontrivial and strongly regular, yet not weakly amenable. The approach hinges on sharp bounds for Körner's functions to enable McKissick-type constructions, combined with Wermer's and Cole's techniques to control the ideal structure and amenability properties. A key achievement is proving $ar{J}_x hd M_x^2$ for all $xelong K$, enabling strong regularity, and then obtaining non-amenable instances both in $R(K)$ and in an essential uniform algebra with bounded relative units. These results settle questions raised by Wilken and Feinstein–Heath, expanding the landscape of possible regularity/amenability configurations in planar uniform algebras and illustrating intricate interactions between local ideals and global algebraic properties.

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.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10
• Corollary 1.11
• Theorem 1.12
• Corollary 1.13