DISJOINTNESS PRESERVING LINEAR OPERATORS BETWEEN BANACH ALGEBRAS OF VECTOR-VALUED FUNCTIONS

We present vector-valued versions of two theorems due to A. Jimenez-Vargas, by showing that, if B(X, E) and B(Y, F) are certain vector-valued Banach algebras of continuous functions and T : B(X, E) -> B(Y, F) is a separating linear operator, then (T) over cap : <(B(X,E))over cap> -><(B(V,F))over cap> defined by (T) over cap(f) over cap =(Tf) over cap, is a weighted composition operator, where c (Tf) over cap is the Gelfand transform of Tf. Furthermore, it is shown that, under some conditions, every bijective separating map T : B(X, E) -> B(Y, F) is biseparating and induces a homeomorphism between the character spaces M(B(X, E)) and M(B(Y, F)). In particular, a complete description of all biseparating, or disjointness preserving linear operators between certain vector-valued Lipschitz algebras is provided. In fact, under certain conditions, if the bijections T : Lip(alpha) (X, E) -> Lip(alpha) (Y, F) and T-1 are both disjointness preserving, then T is a weighted composition operator in the form Tf(y) = h(y)(f(phi(y))), where phi is a homeomorphism from Y onto X and h is a map from Y into the set of all linear bijections from E onto F. Moreover, if T is multiplicative then M(E) and M(F) are homeomorphic.