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

Abstract

It is shown that there exists a compact set $X$ in ${\mathbf{C}}^N$ ($N\ge 2$) such that $\hat{X}\setminus X$ 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 ${\mathbf{C}}^N$ 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.