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

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 $\m{C}_{\m{bd}}(X,k)$ 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 $\m{C}_{\m{bd}}(X,k)$, and non-Archimedean Gel'fand theory.

2010 Mathematics Subject Classification: 11S80, 18B30, 46S10

Keywords and Phrases: Berkovich spectrum, Stone space, Banaschewski compactification, non-Archimedean Gel'fand--Naimark theorem, non-Archimedean Gel'fand theory, non-Archimedean Kaplansky conjecture

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