A survey and new results on Banach algebras of ultrametric continuous functions

Monique Chicourrat (1) , Bertin Diarra (2, 1) , Alain Escassut (2, 1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 LMBP - Laboratoire de Mathématiques Blaise Pascal

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. Maïnetti and the third author concerning the whole algebra A of all bounded continuous functions from IE to IK. We show that every maximal ideal of finite codimension of A is of codimension 1. Moreover, that property 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 spectrum and the Banaschewski compactifiaction of IE is homeomorphic to the multiplicative spectrum of S.

Domaines: Théorie des nombres [math.NT]; Théorie spectrale [math.SP]

Monique Chicourrat, Bertin Diarra, Alain Escassut. A survey and new results on Banach algebras of ultrametric continuous functions. p-Adic Numbers, Ultrametric Analysis and Applications,, In press, 12 (3), pp.185-202. ⟨hal-02882656⟩