Jointly separating maps between vector-valued function spaces

Abstract

AbstractLetXandYbe compact Hausdorff spaces,Ebe a real or complex Banach space andFbe a real or complex locally convex topological vector space. In this paper we study a pair of linear operatorsS,T:A(X,E) →C(Y,F) from a subspaceA(X,E) ofC(X,E) toC(Y,F), which are jointly separating, in the sense thatTfandSghave disjoint cozeros wheneverfandghave disjoint cozeros. We characterize the general form of such maps between certain classes of vector-valued (as well as scalar-valued) spaces of continuous functions including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions. The results can be applied to a pairT:A(X) →C(Y) andS:A(X,E) →C(Y,F) of linear operators, whereA(X) is a regular Banach function algebra onX, such thatf⋅g= 0 impliesTf⋅Sg= 0, for allf∈A(X) andg∈A(X,E). IfTandSare jointly separating bijections between Banach algebras of scalar-valued functions of this class, then they induce a homeomorphism betweenXandYand, furthermore,T−1andS−1are also jointly separating maps.