Characterisation of the Berkovich Spectrum of the Banach Algebra of Bounded Continuous Functions

TL;DR

This paper investigates the structure of the Berkovich spectrum of the Banach algebra of bounded continuous functions over a complete valuation field, providing new insights into non-Archimedean functional analysis.

Contribution

It establishes the universality of the Berkovich spectrum's topological space and applies this to solve problems in non-Archimedean analysis and field extension comparisons.

Findings

Proves universality of the Berkovich spectrum's topology

Provides a partial solution to Kaplansky's conjecture in a non-Archimedean context

Compares ground field extensions of the algebra of bounded continuous functions

Abstract

For a complete valuation field k and a topological space X, we prove the universality of the underlying topological space of the Berkovich spectrum of the Banach k-algebra Cbd(X,k) of bounded continuous k-valued functions on X. This result yields three applications: a partial solution to an analogue of Kaplansky conjecture for the automatic continuity problem over a local field, comparison of two ground field extensions of Cbd(X,k), and non-Archimedean Gel'fand theory.