Classifications of commutative Banach algebras and Banach modules

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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.