Quasianalyticity in certain Banach function algebras

TL;DR

This paper explores quasianalyticity in Banach function algebras defined via a generalized differentiability concept, providing conditions for quasianalyticity and constructing a specific algebra with unique measure properties.

Contribution

It introduces a new notion of differentiability in function algebras and establishes sufficient conditions for quasianalyticity, along with a novel algebra example with specific measure-theoretic properties.

Findings

Established sufficient conditions for quasianalyticity in $\

Constructed a uniform algebra on a locally connected space with no non-trivial Jensen measures that is not regular.

Improved upon a previous example by the first author (2001).

Abstract

Let $X$ be a perfect, compact subset of the complex plane. We consider algebras of those functions on $X$ which satisfy a generalised notion of differentiability, which we call $\mathcal{F}$-differentiability. In particular, we investigate a notion of quasianalyticity under this new notion of differentiability and provide some sufficient conditions for certain algebras to be quasianalytic. We give an application of our results in which we construct an essential, natural uniform algebra $A$ on a locally connected, compact Hausdorff space $X$ such that $A$ admits no non-trivial Jensen measures yet is not regular. This construction improves an example of the first author (2001).