Gleason parts and countably generated closed ideals in \(H^\infty\) (Q2847188)

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)].