Jordan triple endomorphisms and isometries of unitary groups

In this paper we present the general form of all continuous endomorphisms of the group Un of n×n complex unitary matrices with respect to the Jordan triple product. These are the continuous maps ϕ:Un→Un which satisfyϕ(VWV)=ϕ(V)ϕ(W)ϕ(V),V,W∈Un. The result is applied to determine the structure of certain isometries of Un. These include the isometries relative to any metric given by a unitarily invariant norm on the space Mn of all n×n complex matrices and also the isometries relative to any member of a new class of metrics on Un recently introduced by Chau, Li, Poon and Sze [6].