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Survey on the Kakutani problem in p-adic analysis I

Abstract : 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 pointwise convergence, let M ultm(A, .) be the subset of the φ ∈ M ult(A, .) whose kernel is a maximal ideal and let M ult1(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, or ultrametric Kakutani problem the question whether M ult1(A, .) is dense in M ultm(A, .). In order to recall the study of this problem that was made in several successive steps, here we first recall how to characterize the various continuous multiplicative semi-norms of A, with particularly the nice construction of certain multiplicative semi-norms of A whose kernell is neither a null ideal nor a maximal ideal, due to J. Araujo. Here we prove that multbijectivity implies density. The problem of multbijectivity will be described in a further paper.
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Alain Escassut. Survey on the Kakutani problem in p-adic analysis I. Sarajevo Journals of Mathematics, Academy of Sciences and Arts of Bosnia and Herzegovina, 2019, 15 (2), pp.245-263. ⟨hal-02508193⟩

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