Peripheral Multiplicativity of Maps on Uniformly Closed Algebras of Continuous Functions Which Vanish at Infinity

Abstract

We study maps between uniformly closed algebras of complex-valued continuous functions which vanish at infinity on locally compact Hausdorff spaces. Without assuming linearity nor multiplicativity on the maps we show that they are isometrical isomorphisms as Banach space operators if they satisfy that the peripheral range of the product of the images of any two elements coincides with the peripheral range of the product of those elements. Furthermore, if the underlying algebras contain approximate identities, then they are isometrically isomorphic as Banach algebras, which is a generalization of a recent result of Luttman and Tonev for the case of uniform algebras. On the other hand it is not the case without assuming the existence of approximate identities; An example is given.