Douglas algebras on multiply connected domains (Q1086794)

Let \(\Omega\) be a bounded domain in \({\mathbb{C}}\) whose boundary consists of a finite number of disjoint analytic Jordan curves. In this interesting paper it is shown that every closed subalgebra B of \(L^{\infty}=L^{\infty}(\Omega)\) containing \(H^{\infty}=H^{\infty}(\Omega)\) is a Douglas algebra, i.e. is generated by \(H^{\infty}\) and the complex conjugates of single valued interpolating Blaschke products which are invertible in B. This result extends that of \textit{S.-Y. A. Chang} and \textit{D. E. Marshall} for the unit disc [Acta Math. 137, 81-89 and 91-98 (1976; Zbl 0332.46035 and Zbl 0334.46061 resp.)]. A further extension to finite bordered Riemann surfaces is briefly sketched.