Classifications of commutative Banach algebras and Banach modules and its applications

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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 research, paying attention to concrete algebras on locally compact abelian groups, we showed that the set of finite regular Borel measures with natural spectra for a compact abelian group Γ is closed under addition if and only if Γ is discrete. If also G is a non-discrete locally compact abelian group, then there exists a finite reg … More ular Borel measure with natural spectrum of which corresponding multiplier operator on L'(G) is not decomposable. We next proved that certain ring homomorphisms between two commutative Banach algebras with units are automatically linear. Among other things, we proved that ring homomor-phisms on the disk algebra into itself is given in terms including of prime ideals. Moreover Pfaffenberger and Phillips consider a real and unital case of the classical commutative Gelfand theorem and obtain two representation theorems. One is to represent a unital real commutative Banach algebra A as an algebra of continuous functions on the unital homo-morphism space Φ_A. The other is to represent A as an algbera of continuous sections on the maximal ideal space M_A. In this note, we point out that similar theorems for non-unital case hold and show that two representation theorems are essentially identical. We also proved that ring homomorsphisms from real commutative Banach algebras into strictly real commutative Banach algebras are represented by certatain continuous mappings between the maximal ideal spaces.As an application, we considered the following problem, and obtained many valuable results : the Hyers-Ulam stability problem of linear differential operators, a third kind boundary value problem of elliptic differential equation u"= f(t,u) and the correspondence problem between the family of closed convex sets and that of inequalities on a topological vector space. Less