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Bounded analytic functions and closed ideals. (English) Zbl 0633.46055
This paper is a revision of part of the author’s thesis [Ideals and subalgebras of analytic functions (1985; Zbl 0596.46049)]. One of the main results in § 2 is a complete characterization of the closed primary ideals in \(H^{\infty}\) contained in a maximal ideal whose Gleason part is nontrivial. Another result in § 2 tells us that every closed ideal in \(H^{\infty}\) whose hull is contained in the fiber \(M_ 1(H^{\infty})\) of the maximal ideal space of \(H^{\infty}\) has the form \(I=k\cdot J\), where k is the singular inner function \(k(z)=\exp (\alpha (z+1)/(z-1))\) (\(\alpha\geq 0)\) and where J is a closed ideal in \(H^{\infty}\) containing all functions whose Gelfand transforms vanish on \(M_ 1(H^{\infty})\). Extensions of this result may be found in [P. Gorkin, H. Hedenmalm and R. Mortini, Ill. J. Math. 31, 629-644 (1987; Zbl 0615.46049)].
§§ 3 and 4 contain a lot of interesting results relating the structure of closed ideals of Banach algebras B of analytic functions on planar domains W to that of subalgebras A of B which are characterized by the condition that their elements are holomorphically extendable to some of the components of \({\mathbb{C}}\setminus W.\)
The author’s techniques consist in the use of the holomorphic functional calculus. Also, for the first time, the powerful tool of the analytic Carleman transform is successfully applied in the case of nonseparable Banach algebras like \(H^{\infty}(W)\).
Reviewer: R.Mortini

MSC:
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46H30 Functional calculus in topological algebras
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