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arXiv · 2403.19583

A nontrivial uniform algebra Dirichlet on its maximal ideal space

Alexander J. Izzo
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Abstract

It is shown that there exists a nontrivial uniform algebra that is Dirichlet on its maximal ideal space and has a dense set of elements that are exponentials. This answers a 65-year-old question of John Wermer and a 17-year-old question of Garth Dales and Joel Feinstein. Our example is P(X) for a certain compact set X in complex Euclidean 2-space ($\mathbb{C}^2$). It is also shown that there exists a logmodular uniform algebra with proper Shilov boundary but with no nontrivial Gleason parts. This answers a modification of another 65-year-old question of Wermer.

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Alexander J. Izzo. 2024-03-28. A nontrivial uniform algebra Dirichlet on its maximal ideal space. https://arxiv.org/abs/2403.19583

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