Partial regularity and t-analytic sets for Banach function algebras

TL;DR

This paper introduces the concept of t-analytic sets in Banach function algebras, constructs prime ideals, and examines their properties, providing new insights into regularity and conjectures in complex analysis.

Contribution

It defines t-analytic sets, characterizes them for the disk-algebra, and challenges existing assertions about O-analyticity and S-regularity in Banach function algebras.

Findings

Characterization of all t-analytic sets for the disk-algebra

Construction of closed prime ideals using t-analytic sets

Refutation of certain claims regarding O-analyticity and S-regularity

Abstract

In this note we introduce the notion of $t$-analytic sets. Using this concept, we construct a class of closed prime ideals in Banach function algebras and discuss some problems related to Alling's conjecture in $H^\infty$. A description of all closed $t$-analytic sets for the disk-algebra is given. Moreover, we show that some of the assertions in Daoui et al. (Proc. Am. Math. Soc. 131:3211-3220, 2003) concerning the $O$-analyticity and $S$-regularity of certain Banach function algebras are not correct. We also determine the largest set on which a Douglas algebra is pointwise regular.