Topologies on the multiplicative spectrum of an ultrametric Banach algebra: A survey and new results

Let [Formula: see text] be a complete ultrametric field and [Formula: see text] be a [Formula: see text]-Banach algebra. In the past literature the multiplicative spectrum [Formula: see text] of [Formula: see text] has been provided with different topologies and metrics. The usual topology [Formula: see text] is the pointwise convergence topology and it is compact. Actually [Formula: see text] can be provided with some uniform convergence topology. These topologies are uniformizable (in the sense of Bourbaki). Using the uniformity point of view we prove new results about [Formula: see text]. In particular considering the case of a separable [Formula: see text] we get metrization results generalizing some Maïnetti’s results on Tate [Formula: see text]-algebras. Concerning the affine space we give an answer to some question of Maïnetti. We also study the case of some Krasner algebra [Formula: see text] for which a third topology was studied by Boussaf and Maïnetti, this topology is linked to a metric defined on circular filters. Finally, we consider the case of some [Formula: see text]-algebras [Formula: see text] of functions from an ultrametric space [Formula: see text] to [Formula: see text]. Some of these algebras (the admissible and compatible algebras) yet have been studied by the authors. We give here more results relative to uniform properties and compactifications.