On functional analytic approach for corona and Gleason’s problems for holomorphic Lipschitz algebras
DOI:
https://doi.org/10.17398/Keywords:
Corona problem, Gleason’s problem, Gel’fand theory, locally Stein algebras, finitely generated maximal ideal, Lipschitz algebras of holomorphic functions, Gleason A-propertyAbstract
We study Lipschitz algebras of holomorphic functions of the order k, 0 ≤ k ≤ ∞, and the exponent α, α ∈ (0, 1]. The Gel’fand theory and maximal ideal spaces of these algebras are discussed. Further, we solve the corona problem (1962) and Gleason’s problem (1964) for these algebras on certain bounded pseudoconvex/poly domains G in Cn (e.g., the ball and polydisc). As a welcome bonus, we affirmatively solve Fornæss and Øvrelid’s problem (1983) for holomorphic Hölder and Lipschitz spaces. In fact, we establish an equivalency between the two problems for these algebras. As an application, we establish the I.J. Schark’s theorem for Lipschitz algebras on these G’s. Indeed, we extend our recent work on Gleason’s problem, based on the functional analytic approach, as well as extend recent results of Clos for these algebras, and apply the usual Banach algebra method.
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