The cluster value problem in spaces of continuous functions

Abstract

AbstractWe study the cluster value problem for certain Banach algebras ofholomorphic functions defined on the unit ball of a complex Banachspace X. The main results are for spaces of the form X= C(K). 1 Preliminaries. A cluster value problem for a complex Banach space Xis a weak version ofthe corona problem for the open unit ball Bof X, which is a long-standingopen problem in complex analysis when Xhas dimension at least 2. Insteadof studying when B is dense in the spectrum of a uniform algebra H ofbounded analytic functions on Bin the weak topology induced by H(coronaproblem), the cluster value problem investigates the following situation:Let B¯ ∗∗ be the closed unit ball of the bidual X ∗∗ , and let M H be the spec-trum (i.e. maximal ideal space) of a uniform algebra Hof norm continuousfunctions on Bwith H⊃ X ∗ .Then π: M H → B¯ ∗∗ ,given by π(τ) = τ| X ∗ for τ∈ M H ,is surjective (as a consequence of the results in Chapter 2 and5 of [10]). For each x ∗∗ ∈ B¯ ∗∗ ,M x ∗∗ (B) = π −1 (x