Characterisation of the Berkovich spectrum of the Banach algebra of bounded continuous functions
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. NEWLINENEWLINENEWLINE NEWLINEThe first refers to automatic continuity. In classical analysis, i.e. for Banach algebras over the complex field \(\mathbb{C}\), it is well-known that, for any Banach \(\mathbb{C}\)-algebra \(A\), every injective \(\mathbb C\)-algebra homomorphism \(\mathrm{C}_{\mathrm{bd}}(X, \mathbb{C})\rightarrow A\) with closed image is continuous. The author of the present paper provides the \(K\)-version of this result for locally compact \(K\). NEWLINENEWLINENEWLINE NEWLINEThe second refers to ground field extensions. Let \(L/K\) be an extension of complete valuation fields. Then it is proved that the ground field extension \(\mathrm{BSC}_L(X) \rightarrow \mathrm{BSC}_K(X)\) associated with the natural embedding \(\mathrm{C}_{\mathrm{bd}}(X, K) \rightarrow \mathrm{C}_{\mathrm{bd}}(X, L)\) is a homeomorphism. The author also considers the other ground field extension obtained by tensoring \(K\) and studies the difference of those two. NEWLINENEWLINENEWLINE NEWLINEFinally, Theorem 2.1 is applied to establish a Gelfand theory for zero-dimensional Hausdorff spaces \(X\) and locally compact fields \(K\). NEWLINENEWLINENEWLINE NEWLINEMany of the results of this paper were already proved in [\textit{A. Escassut}NEWLINEand \textit{N. Mainetti}, Topology Appl.\ 157, No.~16, 2505--2515 (2010; Zbl 1210.54036)]NEWLINEunder the assumption that \(X\) is an ultrametric space.NEWLINEFor a future research work, it would interesting to investigate whether the results given in this paper for locally compact fields also hold for general complete valuation fields \(K\).
scientific article; zbMATH DE number 6358810