Additive jointly separating maps and ring homomorphisms

Abstract

A(X, E)−→C(Y, F ) is said to be jointly separating if coz(T f ) ∩ coz(Sg) = ∅ whenever coz(f )∩coz(g) = ∅. In this paper, first we give a partial description of additive jointly separating maps between certain spaces of vector-valued continuous functions (including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions). Then we apply the results to characterize continuous ring homomorphisms between certain Banach algebras of vector-valued continuous functions. In particular, the results provide some generalizations of the recent results on unital homomorphisms between vector-valued Lipschitz algebras, with a different approach.