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 $\hat{X}$ such that some point of $\hat{X}\setminus X$ is a one-point Gleason part for $P(X)$. Furthermore, $X$ can chosen so that $P(X)$ has a dense set of invertible elements.

The results of this paper are in a draft of the author's paper arXiv:1801.02252 but have been removed from the latest draft of that paper and given in this new paper at the suggestion of a referee