Regularity points and Jensen measures for $R(X)$

TL;DR

This paper explores the differences between two types of regularity points in Banach function algebras, introduces a new method for constructing Swiss cheese sets with specific properties, and provides examples with positive area of discontinuity points.

Contribution

It introduces a new, more general construction method for Swiss cheese sets where $R(X)$ is non-regular but has no non-trivial Jensen measures, and demonstrates additional properties such as positive area of discontinuity points.

Findings

Regularity points of continuity and R-points can differ in $R(X)$.

New construction method for Swiss cheese sets with specific regularity properties.

Existence of Swiss cheese sets with positive area of discontinuity points.

Abstract

We discuss two types of `regularity point', points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset in an earlier paper on non-regularity for Banach function algebras. We show that, even for the natural uniform algebras $R(X)$ (for compact plane sets X), these two types of regularity point can be different. We then give a new method for constructing Swiss cheese sets $X$ such that $R(X)$ is not regular, but such that $R(X)$ has no non-trivial Jensen measures. The original construction appears in the first author's previous work. Our new approach to constructing such sets is more general, and allows us to obtain additional properties. In particular, we use our construction to give an example of such a Swiss cheese set $X$ with the property that the set of points of discontinuity for $R(X)$ has positive area.