Spectral semi-norm of a {$p$}-adic Banach algebra

Escassut, Alain; Maïnetti, Nicolas

doi:10.36045/bbms/1103408967

Cited by 11 publications

(13 citation statements)

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“…It is well known that the set of maximal ideals is not sufficient to describe spectral properties of an ultrametric Banach algebra: we have to consider the set of continuous multiplicative semi-norms [6], [7], [9], [10], [11], ]. Many studies were made on continuous multiplicative semi-norms on algebras of analytic functions, analytic elements and their applications to holomorphic functional calculus [3], [5], [6]. Here we mean to study continuous multiplicative semi-norms on Banach algebras of continuous functions.…”

Section: Introductionmentioning

confidence: 99%

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Escassut

Maïnetti²

2010

Topology and its Applications

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International audienceLet $K$ be an ultrametric complete field and let $E$ be an ultrametric space. Let $A$ be the Banach $K$-algebra of bounded continuous functions from $E$ to $K$ and let $B$ be the Banach $K$-algebra of bounded uniformly continuous functions from $E$ to $K$. Maximal ideals and continuous multiplicative semi-norms on $A$ (resp. on $B$) are studied by defining relations of stickness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of $A$ (resp. of $B$) is in bijection with the set of equivalence classes with respect to stickness (resp. to contiguousness). Every prime ideal of $A$ or $B$ is included in a unique maximal ideal and every prime closed ideal of $A$ (resp. of $B$) is a maximal ideal, hence every continuous multiplicative semi-norms on $A$ (resp. on $B$) has a kernel that is a maximal ideal. If $K$ is locally compact, every maximal ideal of $A$, (resp. of $B$) is of codimension $1$. Every maximal ideal of $A$ or $B$ 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, on $A$ as on $B$ the set of continuous multiplicative semi-norms defined by points of $E$ is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of $A , \ Max(A)$ and the Banaschewski compactification of $E$ which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of $A$ (resp. $B$) is equal to the whole set of continuous multiplicative semi-norms

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Section: Introductionmentioning

confidence: 99%

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Escassut

Maïnetti²

2010

Topology and its Applications

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“…We will now examine prime closed ideals of D(E). Let us recall the main definitions concerning multiplicative semi-norms [3], [4], [5], [8], [10], [13], [14].…”

Section: Notationmentioning

confidence: 99%

“…Consider a Banach IK-algebra T . It is well known that the set of maximal ideals is not sufficient to describe spectral properties of T : we have to consider the set of continuous multiplicative semi-norms [4], [5], [7], [8], [9], [10], [11], [12], [13], [14]. Many studies were made on continuous multiplicative semi-norms on algebras of analytic functions, analytic elements and their applications to holomorphic functional calculus [1], [3], [4].…”

Section: Introductionmentioning

confidence: 99%

Spectrum of ultrametric Banach algebras of strictly differentiable functions

Escassut

Maïnetti²

2018

Advances in Ultrametric Analysis

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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.

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“…such that Ker(φ) = M but in certain cases, there exist infinitely many φ ∈ M ult m (B, . ) such that Ker(φ) = M [4], [7], [8]. A maximal ideal M of B is said to be univalent if there is only one φ ∈ M ult m (B, . )…”

Section: The Ultrametric Corona Problem By Alain Escassut and Nicolasmentioning

confidence: 99%