2022/11/26 by Alexander J. Izzo, Izzo, Alexander J.
Mathematics · #Advanced Topology and Set Theory #Advanced Topics in Algebra #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2211.14684
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.