Spectrum of Ultrametric Banach Algebras of Strictly Differentiable Functions

Abstract

Let IK be an ultrametric complete field and let E be an open subset of IK of strictly positive codiameter. Let D(E) be the Banach IK-algebra of bounded strictly differentiable functions from E to IK, a notion whose definition is detailed. It is shown that all elements of D(E) have a derivative that is continuous in E. Given a positive number r > 0, all functions that are bounded and are analytic in all open disks of diameter r are strictly differentiable. Maximal ideals and continuous multiplicative semi-norms on D(E) are studied by recalling the relation of contiguity on ultrafilters: an equivalence relation. So, the maximal spectrum of D(E) is in bijection with the set of equivalence classes with respect to contiguity. Every prime ideal of D(E) is included in a unique maximal ideal and every prime closed ideal of D(E) is a maximal ideal, hence every continuous multiplicative semi-norm on D(E) has a kernel that is a maximal ideal. If IK is locally compact, every maximal ideal of D(E) is of codimension 1. Every maximal ideal of D(E) is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently, the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. The Shilov boundary of D(E) is equal to the whole set of continuous multiplicative semi-norms. Many results are similar to those concerning algebras of uniformly continuous functions but some specific proofs are required.

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 .…”