A sharper Swiss cheese
Abstract
It is shown that there exists a compact planar set K such that the uniform algebra R(K) is nontrivial and strongly regular. This settles an issue raised by Donald Wilken 55 years ago. It is shown that the set K can be chosen such that, in addition, R(K) is not weakly amenable. It is also shown that there exists a uniform algebra that has bounded relative units but is not weakly amenable. These results answer questions raised by Joel Feinstein and Matthew Heath 17 years ago. A key ingredient in our proofs is a bound we establish on the functions introduced by Thomas Koerner to simplify Robert McKissick's construction of a nontrivial normal uniform algebra.
Cite
@article{arxiv.2211.14684,
title = {A sharper Swiss cheese},
author = {Alexander J. Izzo},
journal= {arXiv preprint arXiv:2211.14684},
year = {2025}
}
Comments
Many typos have been corrected and many other improvements made. The main results and ideas of the proofs remain unchanged