Prime ideals in closed subalgebras of \(L^{\infty}\) (Q1821312)

Let \({\mathbb{D}}\) denote the open disc and let \(H^{\infty}\) denote the algebra of bounded analytic functions on \({\mathbb{D}}\). The following question was asked by \textit{F. Forelli}: [Linear and complex analysis problem book. Lecture Notes Math. 1043 (1984)]. Let Q be a nonzero prime ideal in \(H^{\infty}\) such that \(Q\neq H^{\infty}\) and suppose Q is (algebraically) finitely generated; do we then have \(Q=\{f\in H^{\infty}:\) \(f(z)=0\}\), where \(z\in {\mathbb{D}}?\) In this paper, it is shown that this question has an affirmative answer. This question was also answered, independently, by \textit{R. Mortini} [Zur Idealstruktur der Disk Algebra A(D) und der Algebra \(H^{\infty}\), Dissertation, Universität Karlsruhe, (1984; Zbl 0545.30039)]. Finitely generated prime ideals in other subalgebras of \(L^{\infty}\) are also studied.