Research on automatic linearities for ring homomorphisms on commutative Banach algebras

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Although it is natural in considering the map on a Banach ring to consider the map(homomorphism) which saves three operations, the ring homomorphism which does not assume preservation of scalar multiplication is sometimes often linear automatically. It considered what case aring homomorphism would turn into an linear map automatically. An representation theorem of the ring homomorphism on a commutative Banach algebra was proved. When the conditions which many commutative Banach algebras fulfill were assumed, it was shown that a ring homomorphism can be represented using the continuation map on the maximum ideal space and the ring homomorphism on a complex number field, and it gave that the ring homomorphism on commutative Banach algebras which contain an element whose spectrum is large using this turned into a linear map. It was shown that especially the ring homomorphism on a disk algebra is complex-linear or conjugate-complex linear, if the non-constant function is contained in the r … More ange. Moreover, a necessary and sufficient condition for a Banach algebra to be of a finite dimension was given using the language of the Banach algebra-valued simultaneous polynomial. Among the composition operators on the Banach algebra of all bounded analytic functions on the unit disk we considered the both cases that a given composition operator is isolated and in an essential component. Moreover, we considered ideals of which closures are intersections of maximal ideals of the Banach algebra of all bounded analytic functions on the disk. It was shown that the ring homomorphism to a real commutative Banach algebra in a narrow sense from a real commutative Banach algebra can be represented by a continuous function of a certain kind between the maximal ideal spaces. Moreover, we considered the division problem. Moreover, we considered the closed ideal generated by one outer function. Moreover, the characterization for absolute norm on complex Euclidian space being smooth was given. The Wirtinger type inequality relevant to an elliptic differential equation and the inequality of Beesack said for relation to be in differential geometry were discussed systematically. It extended to the case of Banach space-valued function. Essential norm and Hyers-Ulam stability constant of a weighted composition operator on C (X) described completely in the language of a certain kind of the same level set. Less