Alling's conjecture on closed prime ideals in \(H^ \infty\)
Let \(H^\infty\) be the algebra of all bounded analytic functions in the open unit disk. At a conference in Oberwolfach around 1970 Norman Alling asked for a characterization of the closed prime ideals in \(H^\infty\). In the paper under review, it is shown that every such ideal is an intersection of maximal ideals, hence uniquely determined by its hull. The proof is based on \textit{L. Carleson's} contour construction developed for the proof of the Corona theorem [Ann. Math., II. Ser. 76, 547-559 (1962; Zbl 0112.29702)], on some recent results of \textit{F. D. Suarez} on separation properties of \(H^\infty\) [J. Funct. Anal. 123, No. 2, 233-263 (1994; Zbl 0808.46076)] and on some factorization theorems of K. Hoffman, K. Izuchi and D. Suarez. Preliminary results on closed prime ideals in \(H^\infty\) have been obtained by the authors in [\textit{R. Mortini}, Bull. Austral. Math. Soc. 35, 213-229 (1987; Zbl 0601.46052)]. Some sufficient conditions for a closed prime ideal to be maximal are given. Alling's conjecture whether the hull of every nonmaximal, closed prime ideal in \(H^\infty\) is the closure of a single Gleason part remains unsolved.
scientific article; zbMATH DE number 1054128
scientific article