Separating maps on weighted function algebras on topological groups

Abstract

Let G 1 and G 2 be locally compact groups and let ω 1 and ω 2 be weight functions on G 1 and G 2, respectively. For i = 1, 2, let also C 0(G i, 1/ω i) be the algebra of all continuous complex-valued functions f on G i such that f/ω i vanish at infinity, and let H: C 0(G 1, 1/ω 1) → C 0(G 2, 1/ω 2) be a separating map; that is, a linear map such that H(f)H(g) = 0 for all f, g ∈ C 0(G 1, 1/ω 1) with fg = 0. In this paper, we study conditions under which H can be represented as a weighted composition map; i.e., H(f) = φ(f ℴ h) for all f ∈ C 0(G 1, 1/ω 1), where φ: G 2 → ℂ is a non-vanishing continuous function and h: G 2 → G 1 is a topological isomorphism. Finally, we offer a statement equivalent to that h is also a group homomorphism.