# Nonmaximal ideals and the Berkovich space of the algebra of bounded analytic functions

## TL;DR

This paper explores the structure of the Berkovich space of bounded analytic functions over nonarchimedean fields, revealing the existence of non-normative seminorms, their kernels, and their relation to the Tate algebra.

## Contribution

It demonstrates the presence of nonmaximal seminorms in the Berkovich space and provides methods to construct families of seminorms with shared kernels, advancing understanding of nonarchimedean analytic geometry.

## Key findings

- Existence of non-normative multiplicative seminorms with nonmaximal kernels
- Methods to generate families of seminorms sharing the same kernel
- Identification of kernels not obtainable by the proposed method

## Abstract

We prove that the Berkovich space of the algebra of bounded analytic functions on the open unit disk of an algebraically closed nonarchimedean field contains multiplicative seminorms that are not norms and whose kernel is not a maximal ideal. We also prove that in general these seminorms are not univocally determined by their kernels, and provide a method for obtaining families of different seminorms sharing the same kernel. On the other hand, we prove that there are also kernels that cannot be obtained by that method. The relation with the Berkovich space of the Tate algebra is also given.