The ultrametric corona problem |
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Authors: | A Escassut N Maïnetti |
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Institution: | (1) Laboratoire de Mathématiques UMR 6620, Université Blaise Pascal, Clermont-Ferrand, Les Cézeaux, 63177 Aubiere Cedex, France;(2) LAIC, EA 2146, IUT, Campus des Cézeaux, Université d’Auvergne, F-63170 Aubiere, France |
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Abstract: | Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk D: |x| < 1. Let Mult(A, ∥ · ∥) be the set of continuous multiplicative semi-norms of A, let Mult
m
(A, ∥ · ∥) be the subset of the ϕ ∈ Mult(A, ∥ · ∥) whose kernel is a maximal ideal and let Mult
a
(A, ∥ · ∥) be the subset of the ϕ ∈ Mult
m
(A, ∥ · ∥) whose kernel is of the form (x − a)A, a ∈ D ( if ϕ ∈ Mult
m
(A, ∥ · ∥) \ Mult
a
(A, ∥ · ∥), the kernel of ϕ is then of infinite codimension). We examine whether Mult
a
(A, ∥ · ∥) is dense inside Mult
m
(A, ∥ · ∥) with respect to the topology of simple convergence. This a first step to the conjecture of density of Mult
a
(A, ∥ · ∥) in the whole set Mult(A, ∥ · ∥): this is the corresponding problem to the well-known complex corona problem. We notice that if ϕ ∈ Mult
m
(A, ∥ · ∥) is defined by an ultrafilter on D, then ϕ lies in the closure of Mult
a
(A, ∥ · ∥). Particularly, we show that this is case when a maximal ideal is the kernel of a unique ϕ ∈ Multm(A, ∥ · ∥). Particularly, when K is strongly valued all maximal ideals enjoy this property. And we can prove this is also true when K is spherically complete, thanks to the ultrametric holomorphic functional calculus. More generally, we show that if ψ ∈ Mult(A, ∥ · ∥) does not define the Gauss norm on polynomials (∥ · ∥), then it is defined by a circular filter, like on rational
functions and analytic elements. As a consequence, if ψ ∈
Multm(A, ∥ · ∥) \ Multa(A, ∥ · ∥) or if φ does not lie in the closure of Mult
a
(A, ∥ · ∥), then its restriction to polynomials is the Gauss norm. The first situation does happen. The second is unlikely.
The text was submitted by the authors in English. |
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Keywords: | Non-Archimedean Corona problem multiplicative semi-norms bounded analytic functions |
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