On characterizations of the image of the Gelfand transform of commutative Banach algebras

In this paper, we propose a new method to characterize in terms of "quasi-topology" the image of Gelfand transform of commutative Banach algebras. Our studies in this paper were inspired by the work of R. Doss in 1967-1968. Among other things, we consider a generalization of Doss' result ([3, Theorem 1] which characterizes Fourier transforms of group algebras) to the case of general commutative Banach algebras. In the last two sections, we study several concrete Banach algebras to see whether our characterization are valid or not; concrete Banach algebras we consider are ideals and quotient algebras of group algebras, Segal algebras for locally compact abelian groups, Lipsitz algebras, C-1([0,1]), C-1 (R) and etc. (C) 2007 Wiley-Vch Verlag G.bH & Co. KGaA, Weinheim.