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 C-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 Cbd(X, and non-Archimedean Gel'fand theory.