Multiplicative spectrum of ultrametric Banach algebras of continuous functions - Archive ouverte HAL Access content directly
Journal Articles Topology and its Applications Year : 2010

## Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Alain Escassut
• Function : Author
• PersonId : 868596

#### Abstract

Let $K$ be an ultrametric complete field and let $E$ be an ultrametric space. Let $A$ be the Banach $K$-algebra of bounded continuous functions from $E$ to $K$ and let $B$ be the Banach $K$-algebra of bounded uniformly continuous functions from $E$ to $K$. Maximal ideals and continuous multiplicative semi-norms on $A$ (resp. on $B$) are studied by defining relations of stickness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of $A$ (resp. of $B$) is in bijection with the set of equivalence classes with respect to stickness (resp. to contiguousness). Every prime ideal of $A$ or $B$ is included in a unique maximal ideal and every prime closed ideal of $A$ (resp. of $B$) is a maximal ideal, hence every continuous multiplicative semi-norms on $A$ (resp. on $B$) has a kernel that is a maximal ideal. If $K$ is locally compact, every maximal ideal of $A$, (resp. of $B$) is of codimension $1$. Every maximal ideal of $A$ or $B$ is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on $E$. Consequently, on $A$ as on $B$ the set of continuous multiplicative semi-norms defined by points of $E$ is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of $A , \ Max(A)$ and the Banaschewski compactification of $E$ which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of $A$ (resp. $B$) is equal to the whole set of continuous multiplicative semi-norms.

#### Domains

Mathematics [math] Number Theory [math.NT]

### Dates and versions

hal-00681916 , version 1 (22-03-2012)

### Identifiers

• HAL Id : hal-00681916 , version 1
• DOI :

### Cite

Alain Escassut, Nicolas Mainetti. Multiplicative spectrum of ultrametric Banach algebras of continuous functions. Topology and its Applications, 2010, 157 (16), pp.2505-2515. ⟨10.1016/j.topol.2010.08.003⟩. ⟨hal-00681916⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

97 View