A nontrivial uniform algebra regular on the Cantor set

TL;DR

This paper constructs a new class of uniform algebras that are logmodular and regular on the Cantor set and other metrizable spaces, including the closed interval, showing they are not necessarily normal.

Contribution

It introduces the first known examples of uniform algebras that are regular on metrizable spaces but not normal, expanding understanding of algebraic regularity properties.

Findings

Existence of nontrivial uniform algebras regular on the Cantor set

Construction of uniform algebras regular on all compact metrizable spaces without isolated points

These algebras are the first known examples not normal despite being regular

Abstract

We prove the existence of a nontrivial uniform algebra that is logmodular and regular on the Cantor set. As a consequence, we obtain that for every compact metrizable space X without isolated points there exists a nontrivial essential uniform algebra that is logmodular and regular on X. In particular, there exists a nontrivial essential uniform algebra that is logmodular and regular on the closed unit interval. Our algebras seem to be the first known uniform algebras that are regular on a metrizable space but are not normal.