QC-level sets and quotients of Douglas algebras
An essentially sup-norm closed subalgebra between \(H^{\infty}\) and \(L^{\infty}\) on the unit circle T is called a Douglas algebra. \(H^{\infty}+C\), where C is the space of continuous functions on T, is the smallest Douglas algebra which contains \(H^{\infty}\) properly. Put \(QC=(H^{\infty}+C)\cap \overline{(H^{\infty}+C)}\), and \(X=M(L^{\infty})\), the maximal ideal space of \(L^{\infty}\). For \(x_ 0\in X\), \(\{x\in X\); \(f(x)=f(x_ 0)\) for every \(f\in QC\}\) is called a QC-level set. For \(f\in L^{\infty}\), put N(f) the closure of the union set of supp \(\mu\) \({}_ x\) such that \(x\in M(H^{\infty}+C)\) and f \(| \sup p \mu_ x\not\in H^{\infty} | \sup p \mu_ x\), where supp \(\mu_ x\) is the support set of the representing measure \(\mu_ x\). Properties of \(N(\bar I)\) for inner functions I, are studied and applied to study QC-level sets and quotients of Douglas algebras. Theorem 1. \(N(\bar I)\) is the smallest weak peak set for \(QC\cap H^{\infty}.\) Theorem 2. There is a Douglas algebra B such that ball \((L^{\infty}/B)\), the closed unit ball of \(L^{\infty}/B\), has extreme points but does not have any exposed points. Theorem 3. There is a Douglas algebra B such that ball \((B/H^{\infty}+C)\) has extreme points but does not have any exposed points. Theorem 4. Under the continuum hypothesis, there is a point x in \(M(H^{\infty}+C)\) such that supp \(\mu_ x\) is a QC-level set.
scientific article; zbMATH DE number 3927666
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