Let Un be a complex unitary group and Dn be its subgroup that consists of n × n diagonal matrices whose diagonal elements are complex numbers of modulus 1. In this paper we present some general results on Jordan triple product maps between Un and Um. As a main result we present the general form of all (continuous) Jordan triple product maps from the group Un to the group Dm. Also, we characterize the general form of all (continuous) maps between the groups Dn and Dm that preserve the Jordan triple product. These are the (continuous) maps ϕ : Dn → Dm which satisfy ϕ(VWV) = ϕ(V)ϕ(W)ϕ(V), V,W ∈ Dn.
