Classifications of commutative Banach algebras and Banach modules and its applications

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In order to understand a subject, one sometimes tries to classify the objects in the subject. By this classification, one may learn the deep aspects of the subject. Thus classification is important for us. Here, classification means setting several conditions and considering the classes satisfying the conditions. In this research, the subject is a commutative Banach algebra or a Banach module, and the purpose is to explain the essence of them. Now we classify them. First we set the natural conditions on them and form the class satisfying the conditions. Next we consider whether the concrete algebra or module belongs to the class. We also investigate the common property of algebras and modules in the class. According to these ideas, we introduce the classes named BSE-algebra and BED-algebra. These are obtained by characterizing the Gelfand transformation image and and Helgason-Wang transformation image of commutative Banach algebras. Then the family of commutative Banach algebras can be … More classified in the following four cases : (I) BSE and BED, (II) BSE and not BED, (III) BED and not BSE, (IV) not BED and not BSE. In this research, we gave the following examples : (I) certain closed idals and quotients of group algebras, commutative C^*-algebras, disk algbera, Hardy algebra, a certain Lipschitz-algebra on the real line; (II) Segal algebras S_P(G), A_P(G) of noncompact LCA group G; (III) L^P-algberas on infinite dimensional compact abelian groups, e^1-algbera on infinite set, C_0(X;τ), A_τ; (IV) the derivation algebra C^1_0 (R) on R, the derivation algebra C^1([0,1]) on [0,1], measure algebras on nondiscrete LCA groups, a certain semigroup algebra. Moreover, we introduced a generalized Segal algbera and investigated functional analysis properties of it. Also we constructed concrete generalized Segal algebras A_<τ(n)>. Furthermore, we constructed the smallest isometrically homeomorphism-invariant Segal algbera in commutative Banach algberas. Especially, in the group algebra case, we characterized its multiplier algbera in terms of a certain local multiplier. As an application, we considered a Hyers-Ulam stability problem of derivation on Banach algebras and functional equations and inequalities on Banach spaces, and obtained many valuable results. Less