Classifications of commutative Banach algebras and Banach modules

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。本文引入了一个新的交换Banach代数群Doss-代数，发展了交换Banach代数的一般分类理论。这种一般的分类理论是基于新引入的概念，称为准拓扑。然后，研究了具体的交换Banach代数是否属于BSE-代数和Doss-代数的上述两个群。此外，委员会认为，我们研究了原模与BSE-模重合的交换Banach代数群以及BSE-模的某个群我们还研究了交换Banach代数的最大正则子代数与Apostol代数重合的群，特别是发现道格拉斯代数属于这样一个群，并且基于Apostol代数有一定的分解自然光谱在研究与交换Banach代数相关的函数空间上的函数时，可以成功地刻画出在非平凡函数空间上的非Lipschitz函数。最后，我们研究了交换Banach代数上的环同态结构，关于Banach范数的不等式和等式，以及BKW-算子。

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 investigation, we have introduced the new group of commutative Banach algebras named Doss-algebras and developed the general classification theory of the commutative Banach algebras. This general classification theory was based on newly introduced concept referred to quasi-topology. Then, whether the concrete commutative Banach algebras belong to the above two groups respective to the BSE-algebras and the Doss-algebras has been investigated. Furthermore, we have studied on the group of commutative Banach algebras such that the original norm coincides with the BSE-norm and on a certain group of BSE-Banach modules.We have also studied on the group of the commutative Banach algebras of which the greatest regular subalgebra coincides with the Apostol algebras and particularly it is found that the Douglas algebra belongs to such a group and has a certain decomposition based on the natural spectra. In the study of the function which operates on the function space related to the commutative Banach algebra, non-Lipschitz functions which operate or do not operate on non-trivial function space can be characterized successfully. Finally, we have investigated a structure of ring-homomorphism on the commutative Banach algebras, inequality and equality with respect to the Banach norm, and a BKW-operator.