Ultrafilters and ultrametric Banach algebras of Lipschitz functions

  • Monique Chicourrat
  • Alain EscassutEmail author
Original Paper


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 \(\lambda _A\).


Ultrametric Banach algebras Ultrafilters Multiplicative spectrum 

Mathematics Subject Classification

46S10 30D35 30G06 



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


  1. 1.
    Berkovich, V.: Spectral theory and analytic geometry over non-archimedean fields. AMS Surv. Monographs 33, 169 (1990)Google Scholar
  2. 2.
    Boussaf, K., Escassut, A.: Absolute values on algebras of analytic elements. Ann. Math. Blaise Pascal 2(2), 15–23 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chicourrat, M., Escassut, A.: Banach algebras of ultrametric Lipschitzian functions. Sarajevo J. Math. 14(27)(2), 1–12 (2018)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Diarra, B., Chicourrat, M., Escassut, A.: Finite codimensional maximal ideals in subalgebras of ultrametric continuous or uniformly continuous functions. Bull. Belg. Math. Soc. 26, 3 (2019) (to appear)Google Scholar
  5. 5.
    Escassut, A., Maïnetti, N.: Shilov boundary for ultrametric algebras, p-adic Numbers in Number Theory. Analytic Geometry and Functional Analysis. Belgian Mathematical Society, Brussels, pp. 81–89 (2002)Google Scholar
  6. 6.
    Escassut, A.: Spectre maximal d’une algèbre de Krasner. Colloq. Math. (Wroclaw) XXXVIII2, 339–357 (1978)CrossRefGoogle Scholar
  7. 7.
    Escassut, A.: The Ultrametric Banach Algebras. World Scientific Publishing Co, Singapore (2003)CrossRefGoogle Scholar
  8. 8.
    Escassut, A., Maïnetti, N.: Multiplicative spectrum of ultrametric Banach algebras of continuous functions. Topology Appl. 157, 2505–2515 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Escassut, A., Maïnetti, N.: Spectrum of ultrametric Banach algebras of strictly differentiable functions. Contemp. Math. 704, 139–160 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Guennebaud, B.: Sur une notion de spectre pour les algèbres normées ultramétriques, thèse d’Etat, Université de Poitiers (1973)Google Scholar
  11. 11.
    Haddad, L.: Sur quelques points de topologie générale. Théorie des nasses et des tramails. Ann. Fac. Sci. Clermont N 44, fasc.7, pp. 3–80 (1972)Google Scholar
  12. 12.
    Samuel, P.: Ultrafilters and compactification of uniform spaces. Trans. Am. Math. Soc. 64, 100–132 (1949)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Van Rooij, A.: Non-archimedean Functional Analysis. Marcel Dekker, New York (1978)zbMATHGoogle Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

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

Personalised recommendations