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Weakly strongly regular uniform algebras

J. F. Feinstein, Alexander J. Izzo

TL;DR

The paper addresses how closed ideals and higher-order derivations behave in uniform algebras, focusing on $R(K)$ for Swiss cheese sets to produce weakly strongly regular but not strongly regular algebras. It develops a square-root construction to amplify Browder-type properties and links higher-order point derivations to the inclusions between closures of powers of maximal ideals. The main result shows that for every $m\ge 2$ there exists a Swiss cheese $K$ containing the origin such that in $R(K)$ we have $\overline{J_x}\supset M_x$ for all $x\neq 0$ and $\overline{J_0}\supset M_0^m$ but not $M_0^{m-1}$, resolving a conjecture of Izzo in a strong form. The work also demonstrates that weakly strongly regular algebras need not admit a global fixed exponent across all points and provides a constructive framework for building such algebras, with broader implications for normality and spectral synthesis-like properties in nonlocal uniform algebras.

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.

Weakly strongly regular uniform algebras

TL;DR

The paper addresses how closed ideals and higher-order derivations behave in uniform algebras, focusing on for Swiss cheese sets to produce weakly strongly regular but not strongly regular algebras. It develops a square-root construction to amplify Browder-type properties and links higher-order point derivations to the inclusions between closures of powers of maximal ideals. The main result shows that for every there exists a Swiss cheese containing the origin such that in we have for all and but not , resolving a conjecture of Izzo in a strong form. The work also demonstrates that weakly strongly regular algebras need not admit a global fixed exponent across all points and provides a constructive framework for building such algebras, with broader implications for normality and spectral synthesis-like properties in nonlocal uniform algebras.

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.
Paper Structure (5 sections, 23 theorems, 25 equations)

This paper contains 5 sections, 23 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Higher-order point derivations and ideals
  3. Square roots of Swiss cheeses
  4. Some preliminary results
  5. Proofs of the main theorems

Key Result

Theorem 1.1

For each integer $m\geq 2$, there exists a Swiss cheese $K$ containing the origin such that $R(K)$ is normal and in $R(K)$ we have $\overline{J_x}\supset M_x$ for every $x\in K\setminus\{0\}$ and $\overline{J_0}\supset M_0^m$ but $\overline{J_0}\not\supset M_0^{m-1}$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 29 more