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

## TL;DR

This paper constructs a specific compact set in complex two-space with a nontrivial polynomial hull, where a point outside the set forms a one-point Gleason part, and the algebra of functions on the set has a dense invertible subset.

## Contribution

It demonstrates the existence of a compact set with a one-point Gleason part outside its boundary and a dense invertible set in its polynomial algebra, advancing understanding of uniform algebras.

## Key findings

- Existence of a compact set with a nontrivial polynomial hull.
- Presence of a one-point Gleason part outside the set.
- Polynomial algebra with a dense set of invertible elements.

## 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.