Research on operating functions on function spaces

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We gave a sufficient conditons for the greatest regular subalgebra and the Apostol algebra (the set of all decomposable multiplication operators) of semi-simple commutative Banach algebras coincide with each other in terms of the maximal ideal spaces. As an application, we investigated stuructures of subalgebras of certain Fourier multipliers which consists of operators with natural spectra. In a special case with typical operating functions, we showed similar phenomenun for certain function spaces. We gave a sufficient condition for the existence of weak projections from commutative C^*-algebras into its subalgebra. We investigated structures of BKW operators on certain function spaces including of the disk algebra. We also characterized BKW operators on the algebara of all real valued continuous functions on the compact intervals under certain additional conditions.We investigated weak products of Blaschke products, and solved a problem of Gorkin and Mortini on prime ideals. We characterized codimension 1 isometries on the Douglas algebras. We characterized the maximal ideal space of commutative C^*-algebra in which every element is the squareof another in case that the maximal ideal space is locally connected. We studied ring homomorphisms on commutative Banach algebras and in the spacial cases, we characterized in terms of mapping on the maximal ideal spaces and ring homomorphisms on the complex number field. As an application of the result we proved automatic linearity results for ring homomorphism on certain semi-simple Commutative Banach algebras. In particular, we proved linearity for ring homomorphisms on the disk algebras whose image contains non-constant functions.