A set in the fiber of \(M(H^ \infty)\) over the point 1 (Q1924927)

Let \(H^\infty\) be the Banach algebra of bounded analytic functions on the open unit disk \(D\) in the complex plane, and let \(M(H^\infty)\) be its maximal ideal space with the weak-star topology. Let \(B\) be the topological boundary of \(D_1\) relative to \(M(H^\infty)\backslash D\), where \(D_1\) is the fiber of \(M(H^\infty)\) over the point 1. Let \(G\) be the set of nontrivial Gleason parts for \(H^\infty\). The main objective of this paper is to show that \(B\) is not a \(G_\delta\) set and \(B\cap G\) is dense in \(B\).