Gleason parts and countably generated closed ideals in \(H^\infty\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1279.30055
DOI: 10.1090/S0002-9947-2013-05693-0
author: Yuko Izuchi
author: Kei-Ji Izuchi
published in: Transactions of the American Mathematical Society
publication date: 2013-09-04T00:00:00Z
Mathematics Subject Classification ID: 30H50
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30J10
zbMATH DE Number: 6205200
zbMATH Keywords: Gleason parts
zbMATH Keywords: countably generated closed ideal
zbMATH Keywords: Carleson-Newman Blaschke product
zbMATH Keywords: algebra of bounded analytic functions
MaRDI profile type: Publication
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cites work: Bounded analytic functions and Gleason parts
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cites work: Factorization of Blaschke products
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cites work: Factorization of Blaschke products and primary ideals in \(H^{\infty }\)
cites work: Factorization of Blaschke products and ideal theory in \(H^{\infty }\)
cites work: Blaschke products satisfying the Carleson-Newman conditions and ideals of the algebra \(H^{\infty}\)
full work available at URL: s0002-9947-2013-05693-0
OpenAlex ID: W2097898921
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title: Gleason parts and countably generated closed ideals in \(H^\infty\)
review text: Let \(M(H^\infty)\) denote the space of nonzero multiplicative functionals on the Banach algebra \(H^\infty\) of bounded analytic functions in the unit disk \(\mathbb D\). Given \(x\in M(H^\infty)\) let NEWLINE\[NEWLINE P(x)=\{y\in M(H^\infty) : \rho(x,y)<1\} NEWLINE\]NEWLINE be its Gleason part (here \(\rho(x,y)=\sup\{ |f(x)| : f(y)=0,\;f\in H^\infty,\;\|f\|_{\infty}\leq 1\}\) denotes the pseudohyperbolic distance). Let \(G\) denote the union of all nontrivial Gleason parts. This paper studies the structure of countably generated ideals \(I\) whose zero set \(Z(I)\) is contained in \(G\). For \(x\in Z(I)\) and \(f\in I\), let \(\text{ord}(f,x)\) denote the order of the zero of \(f\) at \(x\). Let \(m=\sup_{x\in Z(I)} \inf_{f\in I} \text{ord}(f,x)\). The main result (Theorem 1.1) shows that such an ideal is finitely generated if and only if any of the three following conditions holds: NEWLINENEWLINENEWLINENEWLINE (1) There are compact \(\rho\)-separated \(G_\delta\)-subsets \(E_1, \dots, E_m\) of \(G\) such that \(I=\overline{\bigotimes_{j=1}^m I(E_j)}\). A subset \(E\subset M(H^\infty)\) is \(\rho\)-separated if there is \(\epsilon>0\) such that \(\rho(x,y)\geq \epsilon\) for every \(x,y\in E\), \(x\neq y\), and \(I(E)=\{f\in H^\infty : f(x)=0, x\in E\}\). NEWLINENEWLINENEWLINENEWLINE (2) There is a Carleson-Newman Blaschke product \(B\) of order \(m\) in \(I\) such that \(\text{ord}(B,x)=\text{ord}(I,x)\) for every \(x\in Z(I)\), and \(Z(I)\) is a \(G_\delta\)-set. A Blaschke product \(B\) is said to be Carleson-Newman of order \(m\) if \(B=\prod_{j=1}^m b_j\), with \(b_j\) interpolating Blaschke products. NEWLINENEWLINENEWLINENEWLINE (3) There are two Carleson-Newman Blaschke products \(B_1,B_2\) in \(I\) such that \(I=I[B_1,B_2]\). NEWLINENEWLINENEWLINENEWLINE These results extend previous work by both authors [J. Funct. Anal. 259, No. 4, 975--1013 (2010; Zbl 1207.46045), J. Funct. Anal. 260, No. 7, 2086--2147 (2011; Zbl 1250.30049)].
reviewed by: Xavier Massaneda
description: scientific article; zbMATH DE number 6205200