1. ULTRAFILTERS AND ULTRAMETRIC BANACH ALGEBRAS OF LIPSCHITZ FUNCTIONS
- Author
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Alain Escassut, Monique Chicourrat, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)
- Subjects
0209 industrial biotechnology ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Spectrum (functional analysis) ,Order (ring theory) ,02 engineering and technology ,Ultrametric Banach algebras · Ultrafilters · Multiplicative spectrum ,Operator theory ,Lipschitz continuity ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,46S10, 30D35, 30G06 ,020901 industrial engineering & automation ,Bounded function ,Banach algebra ,Shilov boundary ,0101 mathematics ,[MATH]Mathematics [math] ,Ultrametric space ,Analysis ,Mathematics - Abstract
The aim of this paper is to examine Banach algebras of bounded Lipschitz functions from an ultrametric space $$\mathbb {E}$$ to a complete ultrametric field $$\mathbb {K}$$. Considering them as a particular case of what we call C-compatible algebras we study the interactions between their maximal ideals or their multiplicative spectrum and ultrafilters on $$\mathbb {E}$$. We study also their Shilov boundary and topological divisors of zero. Furthermore, we give some conditions on abstract Banach $$\mathbb {K}$$-algebras in order to show that they are algebras of Lipschitz functions on an ultrametric space through a kind of Gelfand transform. Actually, given such an algebra A, its elements can be considered as Lipschitz functions from the set of characters on A provided with some distance $$\lambda _A$$. If A is already the Banach algebra of all bounded Lipschitz functions on a closed subset $$\mathbb {E}$$ of $$\mathbb {K}$$, then the two structures are equivalent and we can compare the original distance defined by the absolute value of $$\mathbb {K}$$, with $$\lambda _A$$.
- Published
- 2020