A normal uniform algebra that fails to be strongly regular at a peak point

TL;DR

This paper constructs a specific normal uniform algebra on a compact space that is not strongly regular at a peak point, answering a long-standing open question and exploring ideal structures within the algebra.

Contribution

It provides the first example of a normal uniform algebra failing strong regularity at a peak point, and develops general results on lifting ideals under Cole root extensions.

Findings

Existence of a normal uniform algebra with a peak point where strong regularity fails.

Construction of a totally ordered family of closed primary ideals with a peak point.

Application of ideal lifting results to create algebras with all points as peak points.

Abstract

It is shown that there exists a normal uniform algebra, on a compact metrizable space, that fails to be strongly regular at some peak point. This answers a 31-year-old question of Joel Feinstein. Our example is R(K) for a certain compact planar set K. Furthermore, it has a totally ordered one-parameter family of closed primary ideals whose hull is a peak point. General results regarding lifting ideals under Cole root extensions are established. These results are applied to obtain a normal uniform algebra, on a compact metrizable space, with every point a peak point but again having a totally ordered one-parameter family of closed primary ideals.