Abstract
Let \({ \rm I\!K}\) be an ultrametric complete valued field and \({ \rm I\!E}\) be an ultrametric space. We examine some Banach algebras \(S\) of bounded continuous functions from \({ \rm I\!E}\) to \({ \rm I\!K}\) with the use of ultrafilters, particularly the relation of stickness. We recall and deepen results obtained in a previous paper by N. Maïnetti and the third author concerning the whole algebra \(\cal A\) of all bounded continuous functions from \({ \rm I\!E}\) to \({ \rm I\!K}\). Every maximal ideal of finite codimension of \(\cal A\) is of codimension \(1\) and we show that this property also holds for every algebra \(S\), provided \({ \rm I\!K}\) is perfect. If \(S\) admits the uniform norm on \({ \rm I\!E}\) as its spectral norm, then every maximal ideal is the kernel of only one multiplicative semi-norm, the Shilov boundary is equal to the whole multiplicative spectrum and the Banaschewski compactification of \({ \rm I\!E}\) is homeomorphic to the multiplicative spectrum of \(S\).
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Chicourrat, M., Diarra, B. & Escassut, A. A Survey and New Results on Banach Algebras of Ultrametric Continuous Functions. P-Adic Num Ultrametr Anal Appl 12, 185–202 (2020). https://doi.org/10.1134/S2070046620030024
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DOI: https://doi.org/10.1134/S2070046620030024