Classifications of commutative Banach algebras and Banach modules

交换 Banach 代数和 Banach 模的分类

• 批准号: 10640150
• 负责人: TAKAHASI Sin-ei
• 金额: $ 1.54万
• 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: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640150/
• 关键词: commutative Banach algebra; BSE-algbera; Segal algebra; Apostol algebra; Fourier multiplier algebra; Gelfand transform; bounded weaka approximate identity; Hlawka type inequalities; ring homomorphism; Hadamard product; Hlawka inequality; Djokov

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, a necessary and sufficient condition for a concrete Segal algebra on a locally compact abelian group to be BSE has been given. We further give a constraction of a minimal bounded weak approximate identities for these algebra. We also show that the greatest regular closed subalgebra and Apostol algebras of semisimple commutative Banach algebras are charaterized in terms of level sets of the maximal ideal spaces. Also the commutative Banach algebra of all Fourier multipliers on the Euclidian space which have have natural spectra has been investigated. Furthermore the relations between all measures with natural spectra on a compact abelian group and Apostol algebra has been considered and it is shown that both do not coincide with each other in the discrete case.Finally, we have investigated a structure of ring-homomorphism on the unital semisimple commutative Banach algebras, a generalization of Mond-Pecauic's theorem on the converse of Jensen's inequality, Hadamard product versions of operator inequalities associated with extensions of Holder- McCarthy-Kantorovich inequality and Hlawka type inequalities on Banach spaces.

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。本文给出了局部紧交换群上的具体Segal代数是BSE的一个充要条件。进一步给出了这类代数的一个极小有界弱逼近恒等式的证明。我们还证明了半单交换Banach代数的最大正则闭子代数和Apostol代数是用极大理想空间的水平集刻画的。并研究了欧氏空间上具有自然谱的Fourier乘子的交换Banach代数。本文还讨论了紧交换群上所有具有自然谱的测度与Apostol代数之间的关系，证明了在离散情形下两者并不一致.最后，我们研究了有单位半单交换Banach代数上的环同态结构，它是Mond-Pecauic关于詹森不等式匡威定理的推广，Banach空间上与保持器- McCarthy-Kantorovich不等式和Hlawka型不等式的推广相关的算子不等式的Hadamard乘积形式

项目成果

Ritsuo Nakamoto: "Generalizations of an inequality of Marcus"Mathematica Japonica. 50-1. 35-39 (1999)

Ritsuo Nakamoto：“马库斯不等式的概括”Mathematica Japonica。

Sin-Ei Takahasi: "A necessary and sufficient condition for equality in the Marcus inequality"Proc.International Conf.Nonlinear Analysis & Convex Analysis 98.. 348-351 (1999)

Sin-Ei Takahasi：“马库斯不等式中平等的充分必要条件”Proc.International Conf.Nonlinear Analysis

Osamu Hatori: "Subalgebras and Fourier multipliers with natural spectra"Comtemporary Mathematics. 232. 171-187 (1999)

Osamu Hatori：“具有自然光谱的子代数和傅立叶乘子”当代数学。

Yuki Seo: "Inequalities of Furuta and Mond-Pecaric on the Hadamard product"Journal of Inequalities and Applications. 5-3. 263-285 (2000)

Yuki Seo：“Furuta 和 Mond-Pecaric 关于 Hadamard 产品的不等式”不等式与应用杂志。

原卓哉: "A weighted extension of Carleman's ineqaulity"京都大学数理解析研究所講究録. 1136. 37-44 (2000)

Takuy​​a Hara：“卡尔曼不等式的加权扩展”京都大学数学科学研究所 Kokyuroku。1136. 37-44 (2000)。