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

• 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

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

分类基于设置若干条件并考虑满足条件的类。本研究的主要目的是澄清交换Banach代数和交换Banach模的本质：首先根据所满足的自然条件对它们进行分类，然后研究具体的代数和模是否属于所分类的群，以及所分类的代数和模具有什么样的不变性质。在此之前，基于上述思想，我们引入并研究了BSE-代数和BSE-Banach模的相应群。本文研究了局部紧交换群上的具体代数，证明了紧交换群Γ的具有自然谱的有限正则Borel测度集在加法下是闭的当且仅当Γ是离散的.如果G也是非离散的局部紧交换群，则存在一个有限的正则群， ...更多信息 具有自然谱的广义Borel测度，其相应的乘子算子在L '（G）上是不可分解的.其次，我们证明了两个具有单位元的交换Banach代数之间的某些环同态是自动线性的。其中，我们证明了圆盘代数上的环同态是以包含素理想的形式给出的。Pfaffenberger和菲利普斯进一步考虑了交换Gelfand定理的一个真实的和酉情形，得到了两个表示定理.一种是将有单位的真实的交换Banach代数A表示为有单位的同态空间Φ A上的连续函数代数。另一种是将A表示为极大理想空间M_A上的连续截面代数。本文指出非酉情形下的类似定理成立，并证明了两个表示定理本质上是相同的。证明了从真实的交换Banach代数到严格真实的交换Banach代数的环同态是由极大理想空间之间的某种连续映射表示的.作为应用，我们考虑了下列问题，得到了许多有价值的结果：线性微分算子Hyers-Ulam稳定性问题，椭圆型微分方程u”= f（t）的第三类边值问题，u）以及拓扑向量空间上闭凸集族与不等式族的对应问题。少

项目成果

Hiroyuki Takagi: "Polynomial identities that imply commutativity of rings"Linear Algbera and its Application. 341. 299-307 (2002)

Hiroyuki Takagi：“暗示环交换性的多项式恒等式”线性代数及其应用。

Osamu Hatori: "Characterizations and automatic linearity for ring homomorphisms on algebras of functions"Contemp.Math.. 328. 201-215 (2003)

Osamu Hatori：“函数代数环同态的特征和自动线性”Contemp.Math.. 328. 201-215 (2003)

Takeshi Miura, Shizuo Miyajima, Sin-Ei Takahasi: "A characterization of Hyers-Ulam stability of first order linear differential operators"J.Math.Anal.Appl.. 286. 136-146 (2003)

Takeshi Miura、Shizuo Miyajima、Sin-Ei Takahasi：“一阶线性微分算子的 Hyers-Ulam 稳定性的表征”J.Math.Anal.Appl.. 286. 136-146 (2003)

高橋眞映, 三浦 毅: "Birtinger型不等式に関する一考察"京都大学数理解析研究所講究録. 1253. 121-123 (2002)

Maye Takahashi、Tsuyoshi Miura：“Birtinger 型不等式的研究”京都大学数学科学研究所 Kokyuroku。1253. 121-123 (2002)

Hiroyuki Takagi, Takeshi Miura, Sin-Ei Takahasi: "The Hyers-Ulam stability of a weighted composition operators on a uniform algebra"J.Nonlinear & Convex Analysis. 5-1. 43-48 (2004)

Hiroyuki Takagi、Takeshi Miura、Sin-Ei Takahasi：“均匀代数上加权合成算子的 Hyers-Ulam 稳定性”J.Nonlinear