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Classifications of commutative Banach algebras and Banach modules and its applications

交换Banach代数和Banach模的分类及其应用

基本信息

批准号：
13640149
负责人：
TAKAHASI Sin-ei
金额：
$ 1.73万
依托单位：
Yamagata University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640149/
关键词：
commutative Banach algebra Gelfand transform locally compact abelian group prime ideal ring homomorphism Hyers-Ulam stability elliptic differential equation Wirtinger-Beesack inequality Banach space maximal ideal weighted composition o

项目摘要

Classifications are based on setting several conditions and considering the classes to satisfy the conditions. This research has been focusing on clarifying the essence of commutative Banach algebras and Banach modules by the following idea : First, they would be classified according to the natural conditions settled, and then whether concrete algebras and modules belong to the classified groups or not, and what invariant properties the specific classified algebra and module have, might be investigated. Before this investigation, based on the above idea we have introduced and investigated the groups respective to BSE-algebras and BSE-Banach modules. In this research, paying attention to concrete algebras on locally compact abelian groups, we showed that the set of finite regular Borel measures with natural spectra for a compact abelian group Γ is closed under addition if and only if Γ is discrete. If also G is a non-discrete locally compact abelian group, then there exists a finite reg … More ular Borel measure with natural spectrum of which corresponding multiplier operator on L'(G) is not decomposable. We next proved that certain ring homomorphisms between two commutative Banach algebras with units are automatically linear. Among other things, we proved that ring homomor-phisms on the disk algebra into itself is given in terms including of prime ideals. Moreover Pfaffenberger and Phillips consider a real and unital case of the classical commutative Gelfand theorem and obtain two representation theorems. One is to represent a unital real commutative Banach algebra A as an algebra of continuous functions on the unital homo-morphism space Φ_A. The other is to represent A as an algbera of continuous sections on the maximal ideal space M_A. In this note, we point out that similar theorems for non-unital case hold and show that two representation theorems are essentially identical. We also proved that ring homomorsphisms from real commutative Banach algebras into strictly real commutative Banach algebras are represented by certatain continuous mappings between the maximal ideal spaces.As an application, we considered the following problem, and obtained many valuable results : the Hyers-Ulam stability problem of linear differential operators, a third kind boundary value problem of elliptic differential equation u"= f(t,u) and the correspondence problem between the family of closed convex sets and that of inequalities on a topological vector space. Less

分类是基于设定几个条件，并考虑满足这些条件的类。本研究的重点是阐明可交换Banach代数和Banach模的本质，其思路是：首先根据确定的自然条件对其进行分类，然后研究具体的代数和模是否属于分类群，以及具体的分类代数和模具有哪些不变性质。在此之前，基于上述思想，我们介绍并研究了bse -代数和BSE-Banach模块对应的群。本文研究了局部紧阿贝尔群上的具体代数，证明了紧阿贝尔群Γ具有自然谱的有限正则Borel测度集当且仅当Γ是离散的，在加法下是封闭的。如果G也是一个非离散的局部紧阿贝尔群，则存在一个具有自然谱的有限正则测度，其相应的乘子算子在L'(G)上不可分解。证明了两个带单位的可交换巴拿赫代数之间的环同态是自动线性的。此外，我们证明了盘代数上的环同态论是用包含素数理想的形式给出的。此外，Pfaffenberger和Phillips还考虑了经典交换Gelfand定理的实酉情形，得到了两个表示定理。一是将一元实交换巴拿赫代数a表示为一元同态空间Φ_A上的连续函数代数。另一种是将A表示为最大理想空间M_A上连续截面的代数。在这篇笔记中，我们指出了非一元情况下的相似定理，并证明了两个表示定理本质上是相同的。证明了从实可交换Banach代数到严格实可交换Banach代数的环同态是用极大理想空间之间的某些连续映射来表示的。作为应用，我们考虑了以下问题，得到了许多有价值的结果：线性微分算子的Hyers-Ulam稳定性问题，椭圆型微分方程u ' = f（t,u）的第三类边值问题，拓扑向量空间上闭凸集族与不等式族的对应问题。少