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Counterexamples to the Gleason problem. (English) Zbl 0933.32002
For a bounded domain \(D\subset\mathbb{C}^n\) let \(R(D)\) be either \(H^\infty (D)\) or \(A(D)\). \(D\) is said to have the \(R\)-Gleason property at \(z_0 \in D\) if the maximal ideal in \(R(D)\) consisting of all the functions vanishing at \(z_0\) is algebraically finitely generated by the coordinate functions \((z_1-z_1^0), \dots, (z_n-z_n^0)\). \(D\) is said to have the \(R\)-Gleason property if it has this property at each \(z_0\in D\).
An example of a pseudoconvex domain \(D\) is constructed that does not have the \(R\)-Gleason property and that has in addition the property of being \(H^\infty\) domain of holomorphy. A sufficient condition for a domain in \(C^n\) to have the \(R\)-Gleason property is also presented.

MSC:
32A17 Special families of functions of several complex variables
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
32D05 Domains of holomorphy
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