Zero product preserving functionals on C(Ω)-valued spaces of functions

Let X be a compact Hausdorff space and Omega be a locally compact sigma-compact space. In this paper we study (real-linear) continuous zero product preserving functionals phi:A?C on certain subalgebras A of the Frechet algebra C(X,C(Omega)). The case that phi is continuous with respect to a specified complete metric on A will also be discussed. In particular, for a compact Hausdorff space K we characterize -continuous linear zero product preserving functionals on the Banach algebra C1([0,1],C(K)) equipped with the norm f=f[0,1]+f '[0,1], where [0,1] denotes the supremum norm. An application of the results is given for continuous ring homomorphisms on such subalgebras.