Gleason Parts and Closed Ideals in Douglas Algebras

Recently, the study of the structure of closed ideals in H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{\infty }$$\end{document} whose zero sets are contained in G, the union set of non-trivial Gleason parts, has progressed remarkably. We generalize these results to closed ideals in Douglas algebras A. For non-zero functions f1,f2,…,fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1,f_2,\ldots ,f_n$$\end{document} in A, I=∑j=1nfjA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I=\sum ^n_{j=1}f_j A$$\end{document} is an ideal (may not be closed) in A. We also show that if I is closed in A and its common zero set is contained in G, then I=bA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I=b A$$\end{document} for a Carleson–Newman Blaschke product b.