A doubly generated uniform algebra with a one-point Gleason part off its Shilov boundary

Abstract

It is shown that there exists a compact set $X$ in ${\mathbb C}^2$ with a nontrivial polynomial hull $\widehat X$ such that some point of $\widehat X \setminus X$ is a one-point Gleason part for $P(X)$. Furthermore, $X$ can chosen so that $P(X)$ has a dense set of invertible elements.