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

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.