Cluster values of analytic functions on a Banach space

Abstract

We investigate uniform algebras of bounded analytic functions on the unit ball of a complex Banach space. We prove several cluster value theorems, relating cluster sets of a function to its range on the fibers of the spectrum of the algebra. These lead to weak versions of the corona theorem for ℓ 2 and for c 0. In the case of the open unit ball of c 0, we solve the corona problem whenever all but one of the functions comprising the corona data are uniformly approximable by polynomials in functions in \({c_0^*}\).