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Multiplicative spectrum of ultrametric Banach algebras of continuous functions

机译：连续函数的超度量Banach代数的乘谱

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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 stickiness 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 stickiness (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.
机译：令K为超度量完整场，令E为超度量空间。设A为从E到K的有界连续函数的Banach K代数，让B为从E到K的有界均匀连续函数的Banach K代数.A上的最大理想和连续乘半范数（分别在B上）是通过定义等效关系的超滤器上的粘性和连续性关系来研究的。因此，A的最大频谱（B的剩余）与关于粘性（等效于连续性）的等价类集合成双射。 A或B的每个素理想都包含在唯一的最大理想中，并且A的每个素封闭理想（代表B的本质）都是最大理想，因此A上的每个连续乘半范数（代表B的个体）都有一个核那是最大的理想。如果K是局部紧致的，则A的每个最大理想值（B的剩余值）都是余维1。A或B的每个最大理想值是唯一连续连续半范数的核，并且每个连续乘法半范数定义为因此，在A上，在B上，由E点定义的连续乘法半范数在所有连续乘法半范数的整个集合中都是密集的。超滤器显示A，Max（A）的连续乘法半范数集与E的Banaschewski压缩之间的双射，这是对连续乘法半范式的拓扑空间同胚的。 A（分别为B）的Shilov边界等于整个连续乘法半范数的集合。

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