Banach Algebras of Ultrametric Lipschitzian Functions

Abstract

We examine Banach algebras of bounded uniformly continuous functions and particularly Lipschitzian functions from an ultrametric space IE to a complete ultrametric field IK: prime and maximal ideals, multiplicative spectrum, Shilov boundary and topological divisors of zero. We get a new compactification of IE similar to the Banaschewski’s one and which is homeomorphic to the multiplicative spectrum. On these algebras, we consider several norms or semi-norms: a norm letting them to be complete, the spectral semi-norm and the norm of uniform convergence (which are weaker), for which prime closed ideals are maximal ideals. When IE is a subset of IK, we also examine algebras of Lipschitzian functions that are derivable or strictly differentiable. Finally, we examine certain abstract Banach IK-algebras in order to show that they are algebras of Lipschitzian functions on an ultrametric space through a kind of Gelfand transform.