Finite codimensional maximal ideals in subalgebras of ultrametric uniformly continuous functions

International audienceLet E be a complete ultrametric space, let K be a perfect complete ultra-metric field and let A be a Banach K-algebra which is either a full K-subalgebra of the algebra of continuous functions from E to K owning all characteristic functions of clopens of E, or a full K-subalgebra of the algebra of uniformly continuous functions from E to K owning all characteristic functions of uniformly open subsets of E. We prove that all maximal ideals of finite codimension of A are of codimension 1. Introduction: Let E be a complete metric space provided with an ultrametric distance δ, let K be a perfect complete ultrametric field and let S be a full K-subalgebra of the K-algebra of continuous (resp. uniformly continuous) functions complete with respect to an ultrametric norm. that makes it a Banach K-algebra [3]. In [2], [4], [5], [6] we studied several examples of Banach K-algebras of functions and showed that for each example, each maximal ideal is defined by ultrafilters [1], [7], [8] and that each maximal ideal of finite codimension is of codimension 1: that holds for continuous functions [4] and for all examples of functions we examine in [2], [5], [6]. Thus, we can ask whether this comes from a more general property of Banach IK-algebras of functions, what we will prove here. Here we must assume that the ground field K is perfect, which makes that hypothesis necessary in all theorems