# Cluster values for algebras of analytic functions

## TL;DR

This paper proves the Cluster Value Theorem for certain algebras of holomorphic functions on Banach spaces with the bounded approximation property, advancing understanding of their spectra and related nullstellensatz results.

## Contribution

It establishes the validity of Cluster Value Theorems for ball and Fréchet algebras on Banach spaces with the bounded approximation property, extending known cases.

## Key findings

- Cluster Value Theorems hold for these algebras under the bounded approximation property.
- Results include weak Nullstellensatz theorems and spectral structure insights.
- Advances previous knowledge limited to trivial cases and Hilbert spaces.

## 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 $A_u(B_X)$, the Banach algebra of all uniformly continuous holomorphic functions on the unit ball $B_X$; and also for the Fr\'echet algebra $H_b(X)$ of holomorphic functions of bounded type on $X$ (more generally, for $H_b(U)$, the algebra of holomorphic functions of bounded type on a given balanced open subset $U \subset X$). 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.