Characterisation of the Berkovich spectrum of the Banach 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 ${\mathrm{C}}_{\mathrm{bd}}(X,k)$ 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 ${\mathrm{C}}_{\mathrm{bd}}(X,k)$, and non-Archimedean Gel'fand theory.