Zero product preserving functionals on $$C(\varOmega )$$-valued spaces of functions

Abstract

Let X be a compact Hausdorff space and $$\varOmega $$ be a locally compact $$\sigma $$ -compact space. In this paper we study (real-linear) continuous zero product preserving functionals $$\varphi : A \longrightarrow {\mathbb {C}}$$ on certain subalgebras A of the Frechet algebra $$C(X,C(\varOmega ))$$ . The case that $$\varphi $$ 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 $$\Vert \cdot \Vert $$ -continuous linear zero product preserving functionals on the Banach algebra $$C^1([0,1],C(K))$$ equipped with the norm $$\Vert f\Vert =\Vert f\Vert _{[0,1]}+\Vert f'\Vert _{[0,1]}$$ , where $$\Vert \cdot \Vert _{[0,1]}$$ denotes the supremum norm. An application of the results is given for continuous ring homomorphisms on such subalgebras.