Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Abstract

International audienceLet $K$ 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

Ultrafilters and ultrametric Banach algebras of Lipschitz functions

“…First, we generalize results obtained in [8] and [9] to some Banach algebras of uniformly continuous bounded functions which we call semi-compatible and C-compatible and which are related to the contiguity relation yet considered in these papers. Actually, considering such an algebra S, we describe interactions between the contiguity relation defined on ultrafilters on IE and maximal ideals or the multiplicative spectrum of S. We prove that the Shilov boundary of S is the multiplicative spectrum itself.…”

“…We will denote by l G(IE) the family of uniformly open subsets of IE. In [8] and [13], dealing with the Banaschewski compactification of IE the authors considered the Boolean ring of clopen sets of IE (with the usual addition ∆ and multiplication ∩). In Section 4 we will consider the Boolean ring of uniformly open sets.…”

“…Except Theorems 2.8 and 2.10 and their corollaries, most of results of this section were given in [8] for the algebra of uniformly continuous functions. Conversely if every element of S admits a limit along U then the mapping χ which associates to each f ∈ S its limit along U is a IK-algebra homomorphism from S to IK admitting M for kernel.…”

Finite codimensional maximal ideals in subalgebras of ultrametric uniformly continuous functions

“…In [2], [4], [5], [6] we studied several examples of Banach IK-algebras of functions and showed that for each example, each maximal ideal is defined by ultrafilters [1], [7], [8] and that each maximal ideal of finite codimension is of codimension 1: that holds for continuous functions [4] and for all examples of functions we examine in [2], [5], [6]. Thus, we can ask whether this comes from a more general property of Banach IK-algebras of functions, what we will prove here.…”

“…Concerning semi-compatible algebras, it just comes from Theorem 3.1 in [2]. Concerning semi-adimissible algebras, we can generalize Theorem 5.1 in [4] and then Corollary 5.2 to any semi-admissible algebras. Thus, we obtain Theorem A:…”