On functional analytic approach for Gleason’s problem in the theory of SCV

Abstract

We establish an equivalency of the Corona problem (1962) and Gleason's problem (1964) in the theory of several complex variables. As an application, we give an affirmative solution of the Corona problem for certain bounded pseudoconvex domains or polydomains in C n including balls and polydiscs. Indeed, we extend our recent work on Gleason's problem based on the functional analytic approach, as well as extend recent work of Clos. We also use this equivalency or else other (functional analytic) methods to affirmatively solve both problems for various Banach spaces of bounded holomorphic functions (including certain holomorphic mixed-norm spaces) on various types of domains in C n such as holomorphic Hölder and Lipschitz spaces (left open by Fornaess and Øvrelid in 1983), holomorphic mean Besov-Lipschitz spaces, Besov-Lipschitz spaces, Hardy-Sobolev spaces, and a weighted Bergman space. The discussion goes via first studying Lipschitz algebras of holomorphic functions of order α, where α ∈ (0, 1]; in particular, the Gel'fand theory and the maximal ideal spaces of these algebras are discussed.[KL1]. Also, we apply the same methods to solve the corona problem for the same familiar Banach spaces of bounded holomorphic functions on various types of domains in C n (including the ball and the polydisc). In the process, we also obtain extensions of certain results in some cases (e.g., see [KL2,K5]). In §5, we give some concluding remarks on our approaches, leading to future directions/open problems. Throughout the paper, "algebra" will mean a complex, commutative algebra with identity.1 History of the two problems and formulation of an equivalency. Corona problem. In 1938, Gel'fand discussed Banach algebras in his thesis. In 1942, Kakutani asked the following fundamental question (known as Corona problem) in the subject: Let H ∞ (D) be the Banach algebra of all bounded, holomorphic functions on the open unit disc D in the complex plane C, equipped with the usual sup-norm, and let f j ∈ H ∞ (D), j = 1, 2, . . . , n, with the property that n j=1 |f j (z)| > ǫ for some positive constant ǫ and all z ∈ D (call it a finite set of corona data on the disc D), do there exist g j ∈H ∞ (D), j = 1, 2, . . . , n, such that n j=1 f j g j = 1 (i.e., n j=1 f j (z)g j (z) = 1

On discontinuity of derivations, inducing inequivalent complete metric topologies

“…Further we recall that in automatic continuity theory, we would normally like to examine when and how the algebraic structure of the algebra A determines the topological structure of A, in particular, the continuity aspect (and more particularly, the uniqueness of the topology of A; see [4,6,12,17,18,20] for more details). So it is natural to expect that the non-uniqueness of the topology would reflect some properties of the algebraic structure of A.…”