Generalized norm preserving maps between subsets of continuous functions

Abstract

Let X and Y be locally compact Hausdorff spaces. In this paper we study surjections $$T: A \longrightarrow B$$ between certain subsets A and B of $$C_0(X)$$ and $$C_0(Y)$$ , respectively, satisfying the norm condition $$\Vert \varphi (Tf, Tg)\Vert _Y=\Vert \varphi (f,g)\Vert _X$$ , $$f,g \in A$$ , for some continuous function $$\varphi : {\mathbb {C}}\times {\mathbb {C}}\longrightarrow {\mathbb {R}}^+$$ . Here $$\Vert \cdot \Vert _X$$ and $$\Vert \cdot \Vert _Y$$ denote the supremum norms on $$C_0(X)$$ and $$C_0(Y)$$ , respectively. We show that if A and B are (positive parts of) subspaces or multiplicative subsets, then T is a composition operator (in modulus) inducing a homeomorphism between strong boundary points of A and B. Our results generalize the recent results concerning multiplicatively norm preserving maps, as well as, norm additive in modulus maps between function algebras to more general cases.