Cluster values for algebras of analytic functions

Abstract

The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space X, we study the Cluster Value Problem for the ball algebra Au(BX), the Banach algebra of all uniformly continuous holomorphic functions on the unit ball BX; and also for the Fréchet algebra Hb(X) of holomorphic functions of bounded type on X (more generally, for Hb(U), the algebra of holomorphic functions of bounded type on a given balanced open subset U⊂X). We show that Cluster Value Theorems hold for all of these algebras whenever the dual of X has the bounded approximation property. These results are an important advance in this problem, since the validity of these theorems was known only for trivial cases (where the spectrum is formed only by evaluation functionals) and for the infinite dimensional Hilbert space. As a consequence, we obtain weak analytic Nullstellensatz theorems and several structural results for the spectrum of these algebras.