One-point Gleason parts and point derivations in uniform algebras

TL;DR

This paper explores the relationship between Gleason parts and point derivations in uniform algebras, demonstrating that these properties can occur independently, with examples of algebras exhibiting one property without the other.

Contribution

It provides new examples showing that uniform algebras can have either nonzero bounded point derivations without nontrivial Gleason parts or vice versa.

Findings

Existence of uniform algebras with nonzero bounded point derivations but no nontrivial Gleason parts

Existence of uniform algebras with nontrivial Gleason parts but no nonzero point derivations

Examples include possibly unbounded point derivations

Abstract

It is shown that a uniform algebra can have a nonzero bounded point derivation while having no nontrivial Gleason parts. Conversely, a uniform algebra can have a nontrivial Gleason part while having no nonzero, even possibly unbounded, point derivations.