Characterisation of the Berkovich spectrum of the Banach algebra of bounded continuous functions
type: Item
instance of: scholarly article
title: Characterisation of the Berkovich spectrum of the Banach algebra of bounded continuous functions
zbMATH Open document ID: 1302.11095
author: Tomoki Mihara
published in: Documenta Mathematica
publication date: 2014-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1303.0397
full work available at URL: 25.html
reviewed by: Cristina Pérez-García
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 46S10
zbMATH DE Number: 6358810
zbMATH Keywords: Berkovich spectrum
zbMATH Keywords: Stone space
zbMATH Keywords: Banaschewski compactification
zbMATH Keywords: non-Archimedean Gelfand-Naimark theorem
zbMATH Keywords: non-Archimedean Gelfand theory
zbMATH Keywords: non-Archimedean Kaplansky conjecture
MaRDI profile type: Publication
arXiv ID: 1303.0397
review text: Non-Achimedean analytic spaces play a central role in various studies in modern number theory. A way to formulate non-Archimedean analytic spaces was given by \textit{V. G. Berkovich} in [Spectral theory and analytic geometry over non-archimedean fields. Providence, RI: American Mathematical Society (1990; Zbl 0715.14013)]. He introduced the spectrum of a Banach algebra over a complete valuation field, which is called the Berkovich spectrum in modern number theory. The Berkovich spectrum of an affinoid algebra is called an affinoid space. Berkovich formulated an analytic space by gluing affinoid spaces with respect to a certain G-topology. NEWLINENEWLINENEWLINE NEWLINELet \(K\) be a complete valuation field, i.e., a field endowed with a valuation of height at most \(1\). In the present paper the author studies the underlying topological space of the Berkovich spectrum \(\mathrm{BSC}_K(X)\) of the Banach algebra \(\mathrm{C}_{\mathrm{bd}}(X, K)\) of bounded continuous \(K\)-valued functions on a topological space \(X\). NEWLINENEWLINENEWLINE NEWLINEThe main result (Theorem 2.1) states that \(\mathrm{BSC}_K(X)\) is naturally homeomorphic to the Stone space \(\mathrm{UF}(X)\) associated to \(X\), which is equipped with the topology constructed using the set of ultrafilters of the Boolean algebra on \(X\) formed by the clopen (i.e., closed and open) subsets of \(X\). This result, together with the fact that \( \mathrm{UF}(X)\) is an initial object in the category of totally disconnected compact Hausdorff spaces under \(X\), lead to conclude that \(\mathrm{BSC}_K(X)\) satisfies the same universality, so it is independent of \(K\). This generalizes a previous result obtained by \textit{B. Banaschewski} in [Math. Nachr. 13, 129--140 (1955; Zbl 0064.41303)] for the particular case of zero-dimensional topological spaces \(X\). NEWLINENEWLINENEWLINE NEWLINEThen the author gives three applications of Theorem 2.1, which connect non-Archimedean analysis and general topology. NEW…
description: scientific article; zbMATH DE number 6358810