The maximal ideal space of algebras of bounded analytic functions on infinitely connected domains

Abstract

Introduction.Let D be a domain in the complex plane, and let H°°(D) be the algebra of bounded analytic functions on D. When equipped with the norm of uniform convergence, H°°(D) becomes a uniform algebra, whose maximal ideal space will be denoted by Ji(D).The domain D can be identified with an open subset of Jt(D), by identifying a point Xe D with the homomorphism "evaluation at A." Each function in H'C(D) has a natural continuous extension on Jt(D), which is given by its Gelfand transform.There are two questions associated with Jt(D) which arise naturally.The first is the corona question, which asks whether D is dense in Jt(D).Carleson [2] proved that the open unit disk A is dense in ^#(A).Stout [9] and others extended Carleson's theorem to finitely connected planar domains, and, more generally, to finite open Riemann surfaces.We will present a technique which gives some information on this count for infinitely connected domains, and which shows that the corona conjecture is true for a certain class of infinitely connected domains.And we will show that if the corona conjecture fails for some domain, then it already fails for domains of particularly simple types, for instance, for a domain obtained from the open unit disk by excising a sequence of disjoint closed subdisks which converge to a prescribed point.The technique involves expressing Hco(D) as essentially a countable direct sum of the algebra //"(A), and invoking Carleson's result.The second question involves determining to what extent an analytic structure can be introduced into subsets of J((D).Define an analytic disk in Jt(D) to be the image of A under any one-to-one map T of A into Jt(D) which has the property that/o Pis analytic for each/e HCC(D).Define an analytic set in J((D) to be any connected set which is a union of analytic disks.Then D is an analytic subset of Jt(D), and it is natural to ask what the other analytic subsets of J((D) are.To approach this problem, the notion of a Gleason part is decisive.Two homomorphisms <p, <p e Ji(D) are said to be in the same Gleason part if sup{W)-0C/)| \feH~(D), ¡/I á 1} < 2.

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