Extreme points of unit balls of quotients of \(L^{\infty}\) by Douglas algebras
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.
scientific article; zbMATH DE number 3919106