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

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 $\widehat X \setminus X$isaone-pointGleasonpartfor$P(X)$\mathrm{.}Furthermore,$X$ can chosen so that $P(X)$ has a dense set of invertible elements.