A nontrivial uniform algebra regular on the Cantor set
J. F. Feinstein, Alexander J. Izzo
TL;DR
The paper constructs a nontrivial uniform algebra on the Cantor set that is logmodular and regular, providing a counterexample to the idea that regularity forces normality on metrizable spaces. By iterating a sophisticated construction and exploiting the Hoffman-Singer fiber algebra, the authors obtain regular, logmodular, non-normal algebras on Cantor-type spaces and extend these to general spaces containing a Cantor set, including the closed unit interval. They further show that essential uniform algebras with these properties exist on every compact metrizable space without isolated points, and use Cole’s root-extension to produce Cantor-set algebras with dense squares. The results illuminate the landscape of regular versus normal uniform algebras on nonconnected, metrizable spaces and introduce techniques that propagate these properties through quotient and extension constructions.
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.