Finite codimensional maximal ideals in subalgebras of ultrametric uniformly continuous functions

Abstract

Let IE be a complete ultrametric space, let IK be a perfect complete ultrametric field and let A be a Banach IK-algebra which is either a full IK-subalgebra of the algebra of continuous functions from IE to IK owning all characteristic functions of clopens of IE, or a full IK-subalgebra of the algebra of uniformly continuous functions from IE to IK owning all characteristic functions of uniformly open subsets of IE. We prove that all maximal ideals of finite codimension of A are of codimension 1.Introduction: Let IE be a complete metric space provided with an ultrametric distance δ, let IK be a perfect complete ultrametric field and let S be a full IK-subalgebra of the IK-algebra of continuous (resp. uniformly continuous) functions complete with respect to an ultrametric norm . that makes it a Banach IK-algebra [3].In [2], [4], [5], [6] we studied several examples of Banach IK-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 IK is perfect, which makes that hypothesis necessary in all theorems.

A Survey and New Results on Banach Algebras of Ultrametric Continuous Functions

“…The main results of this paragraph were allready obtained in [5]. We recall them with all proofs in order to make easy the conclusions of this article.…”

“…We are now able to prove that maximal ideals of finite codimension of S are of codimension 1 in two cases: when S = A and when the field IK is perfect. Theorem 3.3 can be generalized to all semi-admissible algebras provided IK is a perfect field [5].…”

Idempotents in an ultrametric Banach algebra

“…Ultrametric Banach algebras have been a topic of many resarch along the last years [1], [3], [4], [5], [6], [10], [11], [12]. The following Theorem 1.1 (stated in [14]) corresponds in ultrametric Banach algebras to a well known theorem in complex Banach algebra: if the spectrum of maximal ideals admits a partition in two open closed subsets U and V with respect to the Gelfand topology, there exist idempotents u and v such that χ(u) = 1, χ(v) = 0 ∀χ ∈ U and χ(u) = 0, χ(v) = 1 ∀χ ∈ V .…”

Solvability of the p-Adic Analogue of Navier–Stokes Equation via the Wavelet Theory

“…Regarding with diverse applications of the field of p -adic numbers several later papers including applicable contents have been published, e.g., [ 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 ]. These applications in turn motivated the pure mathematicians to develop the new areas of p -adic analysis, containing p -adic wavelet theory (see, e.g., [ 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 ]).…”