Classifications of commutative Banach algebras and Banach modules and its applications

为了理解一个主语，人们有时试图对主语中的宾语进行分类。通过这种分类，人们可以了解这门学科的深层次。因此分类对我们很重要。在这里，分类意味着设置几个条件，并考虑满足条件的类。本研究的主题是可交换的Banach代数或Banach模，目的是解释它们的本质。现在我们对它们进行分类。首先我们给他们设定自然条件，形成满足条件的类。接下来我们考虑具体代数或模块是否属于类。我们也研究了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。这些是通过对交换巴拿赫代数的Gelfand变换象和Helgason-Wang变换象的刻画得到的。则可交换的Banach代数族可分为以下四种情况：(I) BSE与BED， （II） BSE与非BED， （III） BED与非BSE， （IV）非BED与非BSE。本文给出了以下几个例子：(1)群代数、交换C^*-代数、盘代数、Hardy代数、实线上的某一类lipschitz代数的闭理想和商；（II）非紧LCA群G的Segal代数S_P(G), A_P(G)；（III）无穷维紧阿贝尔群上的L^ p -代数，无穷集上的e^1-代数，C_0(X;τ), A_τ；（IV）在R上的导数代数C^1_0 (R)，在[0,1]上的导数代数C^1([0,1])，非离散LCA群上的测度代数，某半群代数。此外，我们引入了广义Segal代数，并研究了它的泛函分析性质。构造了具体的广义Segal代数A_<τ(n)>。进一步构造了交换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来表征它的乘子代数。作为应用，我们研究了Banach代数上的Hyers-Ulam导数稳定性问题以及Banach空间上的泛函方程和不等式，得到了许多有价值的结果。少

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