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Ultrafilters and ultrametric Banach algebras of Lipschitz functions

  • Monique Chicourrat
  • Alain EscassutEmail author
Original Paper

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 \(\lambda _A\).

Keywords

Ultrametric Banach algebras Ultrafilters Multiplicative spectrum 

Mathematics Subject Classification

46S10 30D35 30G06 

Notes

Acknowledgements

The authors are grateful to the anonymous referee who carefully read the paper and pointed out to us useful remarks.

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Université Clermont Auvergne, CNRS, LMBPClermont-FerrandFrance

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