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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：
16540135
负责人：
TAKAHASI Sin-Ei
金额：
$ 2.11万
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：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540135/
关键词：
commutative Banach algebra Gelfand transform locally compact abelian group Segal algbera multiplier homomorphism Hyers-Ulam stability Hua's inequality Segel algebra ideal theory local multiplier approximated identity approximate id

项目摘要

In order to understand a subject, one sometimes tries to classify the objects in the subject. By this classification, one may learn the deep aspects of the subject. Thus classification is important for us. Here, classification means setting several conditions and considering the classes satisfying the conditions. In this research, the subject is a commutative Banach algebra or a Banach module, and the purpose is to explain the essence of them. Now we classify them. First we set the natural conditions on them and form the class satisfying the conditions. Next we consider whether the concrete algebra or module belongs to the class. We also investigate the common property of algebras and modules in the class. According to these ideas, we introduce the classes named BSE-algebra and BED-algebra. These are obtained by characterizing the Gelfand transformation image and and Helgason-Wang transformation image of commutative Banach algebras. Then the family of commutative Banach algebras can be … More classified in the following four cases : (I) BSE and BED, (II) BSE and not BED, (III) BED and not BSE, (IV) not BED and not BSE. In this research, we gave the following examples : (I) certain closed idals and quotients of group algebras, commutative C^*-algebras, disk algbera, Hardy algebra, a certain Lipschitz-algebra on the real line; (II) Segal algebras S_P(G), A_P(G) of noncompact LCA group G; (III) L^P-algberas on infinite dimensional compact abelian groups, e^1-algbera on infinite set, C_0(X;τ), A_τ; (IV) the derivation algebra C^1_0 (R) on R, the derivation algebra C^1([0,1]) on [0,1], measure algebras on nondiscrete LCA groups, a certain semigroup algebra. Moreover, we introduced a generalized Segal algbera and investigated functional analysis properties of it. Also we constructed concrete generalized Segal algebras A_<τ(n)>. Furthermore, we constructed the smallest isometrically homeomorphism-invariant Segal algbera in commutative Banach algberas. Especially, in the group algebra case, we characterized its multiplier algbera in terms of a certain local multiplier. As an application, we considered a Hyers-Ulam stability problem of derivation on Banach algebras and functional equations and inequalities on Banach spaces, and obtained many valuable results. Less

为了理解一个主题，人们有时试图对主题中的对象进行分类。通过这种分类，人们可以了解这个主题的深层方面。因此，分类对我们很重要。这里，分类意味着设置若干条件并考虑满足条件的类。本文以交换Banach代数或Banach模为研究对象，目的是解释它们的本质。现在我们把它们分类。首先，我们给它们设置自然条件，并形成满足这些条件的类。接下来我们考虑具体代数或模是否属于类。我们还研究了这类代数和模的共同性质。根据这些思想，我们引入了BSE-代数和BED-代数。通过刻画交换Banach代数的Gelfand变换象和Helgason-Wang变换象得到了这些结果。那么交换Banach代数族可以是 ...更多信息分类为以下四种情况：（I）BSE和BED，（II）BSE和非BED，（III）BED和非BSE，（IV）非BED和非BSE。在本研究中，我们给出了以下例子：（I）群代数，交换C^*-代数，圆盘代数，哈代代数，真实的直线上的Lipschitz-代数的某些闭恒等式和闭代数;（II）非紧LCA群G的Segal代数S_P（G），A_P（G）; τ），A_τ;（IV）R上的导子代数C^1_0（R），[0，1]上的导子代数C^1（[0，1]），非离散LCA群上的测度代数，某个半群代数.引入了广义Segal代数，研究了它的泛函分析性质，并构造了具体的广义Segal代数A <τ（n）>.在此基础上，构造了交换Banach代数中最小的等距同胚不变Segal代数.特别地，在群代数的情况下，我们用某个局部乘子刻画了它的乘子代数a。作为应用，我们考虑了Banach代数上导子的Hyers-Ulam稳定性问题和Banach空间上函数方程和不等式的Hyers-Ulam稳定性问题，得到了许多有价值的结果。少