The cluster value problem for Banach spaces

TL;DR

This paper reduces the cluster value problem for certain Banach algebras in separable Banach spaces to a simpler case involving $ ext{ell}_1$ sums of finite-dimensional spaces, advancing understanding in functional analysis.

Contribution

It establishes a reduction of the cluster value problem in separable Banach spaces to $ ext{ell}_1$ sums of finite-dimensional spaces, providing a new approach to this problem.

Findings

Reduction of the cluster value problem to $ ext{ell}_1$ sums of finite-dimensional spaces

Applicable to Banach algebras $A_u$ and $H^{inity}$ in separable spaces

Advances the understanding of the cluster value problem in Banach space theory

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.