Regularity and amenability conditions for uniform algebras

TL;DR

This paper surveys the relationship between regularity and amenability in uniform algebras, demonstrating that certain regularity conditions do not necessarily imply weak amenability through constructed counterexamples.

Contribution

It provides new insights into the connection between regularity conditions and amenability, including a counterexample showing these properties are not always aligned.

Findings

Bounded relative units imply density of locally constant functions in separable uniform algebras.

Existence of a regular uniform algebra with all points as peak points that is not weakly amenable.

Continuous derivations may not annihilate the set of locally constant functions in certain uniform algebras.

Abstract

We give a survey of the known connections between regularity conditions and amenability conditions in the setting of uniform algebras. For a uniform algebra $A$ we consider the set, $A_{lc}$, of functions in $A$ which are locally constant on a (varying) dense open subset of the character space of $A$. We show that, for a separable uniform algebra $A$, if $A$ has bounded relative units at every point of a dense subset of the character space of $A$, then $A_{lc}$ is dense in $A$. We construct a separable, essential, regular uniform algebra $A$ on its character space $X$ such that every point of $X$ is a peak point for $A$, $A$ has bounded relative units at every point of a dense open subset of $X$ and yet $A$ is not weakly amenable. In particular, this shows that a continuous derivation from a separable, essential uniform algebra $A$ to its dual need not annihilate $A_{lc}$.