Abstract
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We show that there exists a normal uniform algebra, on a compact
metrizable space, that fails to be strongly regular at a peak point.
This answers a 32-year-old question of Joel Feinstein. Our example is
for a certain
compact planar set
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Furthermore, our example has a totally ordered one-parameter family of closed
primary ideals whose hull is a peak point. We establish general results regarding
lifting ideals under Cole root extensions. 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.
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Keywords
normal uniform algebra, strongly regular, bounded relative
units, peak point, primary ideal, point derivation, root
extension
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Mathematical Subject Classification
Primary: 30H50, 46J10, 46J15
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Milestones
Received: 27 April 2024
Accepted: 10 August 2024
Published: 2 October 2024
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| © 2024 The Author(s), under
exclusive license to MSP (Mathematical Sciences Publishers).
Distributed under the Creative Commons
Attribution License 4.0 (CC BY). |
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