Analytic properties in the spectrum of certain Banach algebras (Q957925)

Let \(D\subset \mathbb{C}^n\) be a bounded pseudoconvex domain and let \(S(bD)\) denote the set of strictly pseudoconvex boundary points. By \(\mathcal{B}(D)\), the author denotes one of the Banach algebras \(H^\infty (D)\) or \(A(D)\) or \(A^k(D).\) The domain \(D\) is said to have the Gleason \(\mathcal{B}\) property at \(p\in D\) if for every \(f\in \mathcal{B}(D)\) such that \(f(p)=0\), there exist functions \(f_1,\dots , f_n\in \mathcal{B}(D)\) such that \(f(z)=\sum (z_j-p_j)f_j(z)\) for all \(z\in D.\) Now let \(V \subset \subset S(bD)\cap C^2\) be a non-empty open set in the intersection of \(bD\) and the convex hull of \(D.\) It is shown that for every point \(\xi \in V\), there exists an open neighborhood \(U_\xi\) of \(\xi\) such that \(D\) has the Gleason \(\mathcal{B}\) property at \(p\) for all \(p\in U_\xi \cap D.\) In the sequel, the author discusses some properties concerning spectrum schlichtness following the approach of [\textit{U. Backlund} and \textit{A. Fällström}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 26, No.~3, 595--603 (1998; Zbl 0933.32002)].