A Hull with No Nontrivial Gleason Parts

TL;DR

This paper constructs a polynomially convex hull with all points as single Gleason parts and no bounded point derivations, advancing understanding of hull structures and counterexamples in complex analysis.

Contribution

It introduces a hull with all points as one-point Gleason parts and no bounded point derivations, strengthening Stolzenberg's result and providing a counterexample to the peak point conjecture.

Findings

Existence of a polynomially convex hull with only one-point Gleason parts

Hull contains no nonzero bounded point derivations

Provides a counterexample to the peak point conjecture

Abstract

The existence of a nontrivial polynomially convex hull with every point a one-point Gleason part and with no nonzero bounded point derivations is established. This strengthens the \hbox{celebrated} result of Stolzenberg that there exists a nontrivial polynomially convex hull that contains no analytic discs. A doubly generated counterexample to the peak point conjecture is also presented.