Extreme points of unit balls of quotients of \(L^{\infty}\) by Douglas algebras
type: Item
instance of: scholarly article
title: Extreme points of unit balls of quotients of \(L^{\infty}\) by Douglas algebras
zbMATH Open document ID: 0575.46048
published in: Illinois Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: An essentially sup-norm closed subalgebra between \(H^{\infty}\) and \(L^{\infty}\) on the unit circle T is called a Douglas algebra. For a peak subset E of \(\partial H^{\infty}\), the Shilov boundary of \(H^{\infty}\), put \(H_ E^{\infty}=\{f\in L^{\infty}\); \(f|_ E\in H^{\infty}|_ E\}\), then \(H_ E^{\infty}\) is a Douglas algebra. \textit{D. H. Luecking} and \textit{R. M. Younis} [Trans. Am. Math. Soc. 276, 699-709 (1983; Zbl 0514.30037)] showed that \(ball(L^{\infty}/H_ E^{\infty})\), the closed unit ball of \(L^{\infty}/H_ E^{\infty}\), does not have extreme points. The following theorem is its generalization. Theorem. Let B be a Douglas algebra with \(H^{\infty}\subsetneqq B\). If the essential set of B contains a closed \(G_{\delta}\) subset of \(\partial H^{\infty}\), then \(ball(L^{\infty}/B)\) does not have extreme points. Here the essential set of B is the smallest subset of \(\partial H^{\infty}\) such that every function f in \(L^{\infty}\) which vanishes on it belongs to B.
Mathematics Subject Classification ID: 46J30
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 3919106
zbMATH Keywords: quotient space
zbMATH Keywords: Douglas algebra
zbMATH Keywords: peak subset
zbMATH Keywords: Shilov boundary
zbMATH Keywords: extreme points
zbMATH Keywords: essential set
author: Kei-Ji Izuchi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3919106