Maximality theorems for some closed subalgebras between A and \(H^{\infty}\) (Q1085399)

The authors consider a pair of function algebras \({\mathcal A}\), \({\mathcal C}\) for which \(C\subseteq {\mathcal C}\subseteq L^{\infty}\) and \(A\subseteq {\mathcal A}\subseteq H^{\infty}\), where C denotes the continuous functions on the unit circle T and A is the disk algebra. Criteria are obtained for \({\mathcal A}\) to be ''quasimaximal'', i.e., to have the property that if D is a closed subalgebra of C for which \(D\varsupsetneq {\mathcal A}\), then \(D\supseteq C\). These criteria concern the uniqueness of a representing measure for the complex homomorphism \(\phi_ 0(f)=(1/2\pi)\int^{2\pi}_{0}f(e^{i\theta})d\theta\) on \({\mathcal A}\), and the condition that z\({\mathcal A}=\{f\in {\mathcal A}:\phi_ 0(f)=0\}\). The authors conclude by applying these results to several specific examples.