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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 64-year-old question of John Wermer and a 16-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 (C²). It is also shown that there exists a logmodular uniform algebra with proper Shilov boundary but with no nontrivial Gleason parts. This answers a modified form of another 64-year-old question of Wermer.
Alexander J. Izzo (2024) studied this question.