2025/01/27 by Feinstein, J. F., Izzo, Alexander J.
#30H50 #46J10 #46J15 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2501.15750
Given a uniform algebra A on a compact Hausdorff space X and a point x in X, denote by Mx the ideal of functions in A that vanish at x and by Jx 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 Jx contains Mx for every x in K minus 0 and the closure of J0 contains M0m but does not contain M0m-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.