About the ultrametric corona problem

Escassut, Alain; Maïnetti, Nicolas

doi:10.1016/j.bulsci.2007.09.002

Cited by 8 publications

(26 citation statements)

References 15 publications

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“…Concerning uniformly continuous functions, it has been shown that two ultrafilters that are not contiguous define two distinct continuous multiplicative seminorms. Now, concerning bounded analytic functions inside the disk F = {x ∈ K |x| < 1}, in [9], it was shown that the same property holds for a large set of ultrafilters on F . However, the question remains whether it holds for all ultrafilters on F .…”

Section: Remark 15mentioning

confidence: 91%

“…|. 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].…”

Section: Introductionmentioning

confidence: 99%

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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: Remark 15mentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Escassut

Maïnetti²

2010

Topology and its Applications

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“…Two ultrafilters U, V are said to be contiguous if for every > 0, there exists X ∈ U and Y ∈ V such that the distance from X to Y is less than . Then, as noticed in [10], two contiguous coroner ultrafilters define the same ideal. Conversely, if two coroner ultrafilters U, V define the same ideal, are they contiguous?…”

Section: We Know That Sup{φmentioning

confidence: 80%

“…A similar topology exists on a Banach IKalgebra when all maximal ideals have codimension 1. But as explained in [10], this makes no sense when certain maximal ideals are of infinite codimension, which is the case for our algebra A, since the maximal ideals which are not of the form (x − a)A are of infinite codimension [10] and therefore, there is no Gelfand topology on the whole set of maximal ideals of A. Consequently, a Corona problem should be defined in a different way, as explained in [10]. However, in [19] a "Corona Statement" similar to that mentioned above was shown in our algebra A and it is useful in the present paper as it was in [10].…”

Section: Introduction and Resultsmentioning

confidence: 98%

The Corona problem on a complete ultrametric algebraically closed field

Escassut

2016

P-Adic Num Ultrametr Anal Appl

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Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the "open" unit disk D of IK provided with the Gauss norm. Let M ult(A, .) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let M ult m (A, .) be the subset of the φ ∈ M ult(A, .) whose kernel is a maximal ideal and let M ult 1 (A, .) be the subset of the φ ∈ M ult(A, .) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem the question whether M ult 1 (A, .) is dense in M ult m (A, .). In a previous paper, it was proved that when IK is spherically complete, the answer is yes. Here we generalize this result to any algebraically closed complete ultrametric field, which particularly applies to l C p. On the other hand, we also show that the continuous multiplicative semi-norms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of M ult 1 (A, .), which suggest that M ult 1 (A, .) might be dense in M ult(A, .).

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“…Points of type I can be identified with those in D (see [8]), as each of them is the absolute value evaluation δ z at a point z of D (that is, δ z (f ) = |f (z)| for every f ∈ H ∞ ).…”

Section: Introductionmentioning

confidence: 99%

Nonmaximal ideals and the Berkovich space of the algebra of bounded analytic functions

Araujo

2017

Journal of Mathematical Analysis and Applications

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Abstract. We prove that the Berkovich space of the algebra of bounded analytic functions on the open unit disk of an algebraically closed nonarchimedean field contains multiplicative seminorms that are not norms and whose kernel is not a maximal ideal. We also prove that in general these seminorms are not univocally determined by their kernels, and provide a method for obtaining families of different seminorms sharing the same kernel. On the other hand, we prove that there are also kernels that cannot be obtained by that method. The relation with the Berkovich space of the Tate algebra is also given.

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About the ultrametric corona problem

Cited by 8 publications

References 15 publications

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

The Corona problem on a complete ultrametric algebraically closed field

Nonmaximal ideals and the Berkovich space of the algebra of bounded analytic functions

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