Gelfand-Kolmogoroff theorem for rings of analytic functions

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• 作者：Bedanta Bose^a ; ^{ana_bedanta@yahoo.com} ; Mayukh Mukherjee^b ; ^{mukherjee@mpim-bonn.mpg.de}
• 关键词：54D35 ; 30H50
• 刊名：Topology and its Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：218
• 期：Complete
• 页码：53-65
• 全文大小：403 K
• 卷排序：218

文摘

Consider rings of single variable real analytic or complex entire functions, denoted by K〈z〉K〈z〉. We study “discrete z  -filters” on KK and their connections with the space of maximal ideals of K〈z〉K〈z〉. We characterize the latter as a compact T1T1 space θKθK of discrete z  -ultrafilters on KK. We show that θKθK is a bijective continuous image of βK∖Q(K)βK∖Q(K), where Q(K)Q(K) is the set of far points of βKβK. θKθK turns out to be the Wallman compactification of the canonically embedded image of KK inside θKθK. Using our characterization of θKθK, we derive a Gelfand–Kolmogoroff characterization of maximal ideals of K〈z〉K〈z〉 and show that the Krull dimension of K〈z〉K〈z〉 is at least c. We also establish the existence of a chain of prime z  -filters on KK consisting of at least 2c2c many elements.