Classifications of commutative Banach algebras and Banach modules and its applications

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I n 1990, the author and the cooperation researchers, Prof. O. Hatori, introduced the class called BSE-algebras. In the present study, the author has successfully classified commutative Banach algebras by using the new concept that is called quasi-topology originating from BSE-algebras. Especially, with this classification, the author made clear the position of important commutative Banach algebras called Segal algebras. In addition, as applications of this classification, certain Ulam type stability problems are solved and several inequalities are derived for convex functions.