Continuous maps preserving Jordan triple products from GLn(C) into C∗

Let GLn(C) be a complex general linear group and DGLn(C) be its subgroup comprising n×n diagonal matrices in which the diagonal elements are non-zero complex numbers. In this paper we present some general results on Jordan triple product maps between GLn(C) and GLm(C). The result is applied to determine the structure of Jordan triple product maps between GLn(C) and GL1(C)≅C∗. As a main result we present the general form of all continuous Jordan triple product maps from the group GLn(C) to the group C∗. Also, we characterize the general form of all continuous maps between the groups DGLn(C) and DGLm(C) that preserve the Jordan triple product. These are the continuous maps φ:DGLn(C)→DGLm(C) which satisfy φ(VWV)=φ(V)φ(W)φ(V),V,W∈DGLn(C).