Weakly strongly regular uniform algebras
Complex Variables
2025-10-20 v2 Functional Analysis
Abstract
Given a uniform algebra A on a compact Hausdorff space X and a point x in X, denote by M_x the ideal of functions in A that vanish at x and by J_x the ideal of functions in A that vanish on a neighborhood of x. It is shown that for each integer m greater than or equal to 2, there exists a compact plane set K containing the origin such that in R(K) the closure of J_x contains M_x for every x in K minus {0} and the closure of J_0 contains M_0^m but does not contain M_0^{m-1}. This result establishes a recent conjecture of Alexander Izzo. For the proof we introduce a construction that could be described as taking square roots of Swiss cheeses.
Cite
@article{arxiv.2501.15750,
title = {Weakly strongly regular uniform algebras},
author = {J. F. Feinstein and Alexander J. Izzo},
journal= {arXiv preprint arXiv:2501.15750},
year = {2025}
}
Comments
Added a Corollary (Corollary 1.2) and made some additional minor changes