A Survey and New Results on Banach Algebras of Ultrametric Continuous Functions

Let IK be an ultrametric complete valued field and IE be an ultrametric space. We examine some Banach algebras S of bounded continuous functions from IE to IK with the use of ultrafilters, particularly the relation of stickness. We recall and deepen results obtained in a previous paper by N. Mainetti and the third author concerning the whole algebra A of all bounded continuous functions from IE to IK. Every maximal ideal of finite codimension of A is of codimension 1 and we show that this property also holds for every algebra S, provided IK is perfect. If S admits the uniform norm on IE as its spectral norm, then every maximal ideal is the kernel of only one multiplicative semi-norm, the Shilov boundary is equal to the whole multiplicative spectrumand the Banaschewski compactification of IE is homeomorphic to the multiplicative spectrum of S.