Separating maps on weighted functions algebras on topological groups

Separating maps on weighted function algebras on topological groups

Saeid Maghsoudi

• Department of Mathematics, University of Zanjan, Zanjan, 45195-313, Iran

Rasoul Nasr-Isfahani

• Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 841546-83111, Iran

• School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box: 19395-5746, Tehran, Iran

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.

Keywords: convolution quasi-homomorphism, locally compact group, separating map, weight function, weighted function algebras.

MSC: Primary 43A15, 46J10, Secondary 47B38.