Gleason parts and point derivations for uniform algebras with dense invertible group

TL;DR

This paper constructs specific compact sets in complex spaces where the associated uniform algebra exhibits dense invertible elements, large Gleason parts, and abundant point derivations, addressing longstanding questions in complex analysis.

Contribution

It demonstrates the existence of compact sets with dense invertible groups and large Gleason parts in uniform algebras, extending results to higher dimensions and rational hulls.

Findings

Existence of compact sets with dense invertible elements and large Gleason parts.

Construction of Swiss cheese sets with full measure Gleason parts and point derivations.

Analogous results for rational hulls and polynomial hulls without analytic discs.

Abstract

It is shown that there exists a compact set $X$ in ${\bf C}^N$ ($N\geq 2$) such that $\widehat X\setminus X$ is nonempty and the uniform algebra $P(X)$ has a dense set of invertible elements, a large Gleason part, and an abundance of nonzero bounded point derivations. The existence of a Swiss cheese $X$ such that $R(X)$ has a Gleason part of full planar measure and a nonzero bounded point derivation at almost every point is established. An analogous result in ${\bf C}^N$ is presented. The analogue for rational hulls of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is established. The results presented address questions raised by Dales and Feinstein.