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- Title
SURVEY ON THE KAKUTANI PROBLEM IN P-ADIC ANALYSIS I.
- Authors
ESCASSUT, ALAIN
- 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 Mult(...) be the set of continuous multiplicative semi-norms of A provided with the topology of pointwise convergence, let Multm(...) be the subset of the ε Mult(...) whose kernel is a maximal ideal and let Mult1(...) be the subset of the φ ε Mult(...) whose kernel is a maximal ideal of the form (x-a)A with a ε D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem, or ultrametric Kakutani problem the question whether Mult1(...) is dense in Multm(...). 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.
- Subjects
P-adic analysis; SET theory; MATHEMATICS; MATHEMATICAL analysis; TOPOLOGY
- Publication
Sarajevo Journal of Mathematics, 2019, Vol 15, Issue 2, p245
- ISSN
1840-0655
- Publication type
Article
- DOI
10.5644/SJM.15.02.09