The cluster value problem for Banach spaces

Abstract

The main result is that the cluster value problem in separable Banach spaces, for the Banach algebras $A_{u}$ and $H^{\infty}$, can be reduced to the cluster value problem in those spaces which are $\ell_{1}$ sums of a sequence of finite dimensional spaces. In particular, we prove that the cluster value problem for $\ell_{1}$ is equivalent to the cluster value problem for $L_{1}(0,1)$.