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arXiv · 2501.15750

Weakly strongly regular uniform algebras

J. F. Feinstein·Alexander J. Izzo
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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.

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J. F. Feinstein, Alexander J. Izzo. 2025-01-27. Weakly strongly regular uniform algebras. https://arxiv.org/abs/2501.15750

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