Classifications of commutative Banach algebras and Banach modules

分类的基础是设置几个条件，并考虑满足条件的类。本研究一直致力于阐明交换Banach代数和Banach模的本质：首先，根据给定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分类群，以及特定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，我们引入并研究了BSE-代数和BSE-Banach模各自的群。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。我们进一步给出了这些代数的一个极小有界弱近似恒等式的构造。利用极大理想空间的水平集刻画了半单交换Banach代数的最大正则闭子代数和Apostol代数。还研究了欧氏空间上所有有自然谱的傅里叶乘子的交换Banach代数。最后，我们研究了单位半单交换Banach代数上的一种环同态结构，推广了Jensen不等式的逆的Mond-Pecauic定理，推广了与Holder-McCarthy-Kantorovich的推广有关的算子不等式的Hadamard乘积形式，推广了Banach空间上的Hlawka型不等式。

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.