Zero product preserving maps of operator-valued functions

Let X, Y be locally compact Hausdorff spaces and M, N be Banach algebras. Let theta : C-0(X,M) --> C-0(Y,N) be a zero product preserving bounded linear map with dense range. We show that theta is given by a continuous field of algebra homomorphisms from M into N if N is irreducible. As corollaries, such a subjective theta arises from an algebra homomorphism, provided that M is a W*-algebra and N is a semi-simple Banach algebra, or both M and N are C*-algebras.