Ultrafilters and ultrametric Banach algebras of Lipschitz functions

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 $$\lambda _A$$.