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Journal Articles Contemporary mathematics Year : 2018

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 differen-tiable. 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. Introduction and preliminaries:
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Dates and versions

hal-01918238 , version 1 (10-11-2018)

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Cite

Alain Escassut, Nicolas Maïnetti. Spectrum of ultrametric Banach algebras of strictly differentiable functions. Contemporary mathematics, 2018, Advances in Ultrametric Analysis, 704, pp.139-159. ⟨10.1090/conm/704/14165⟩. ⟨hal-01918238⟩
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