Jordan triple endomorphisms and isometries of spaces of positive definite matrices

Abstract

In this paper, we determine the structure of certain algebraic morphisms and isometries of the space of all complex positive definite matrices. In the case , we describe all continuous Jordan triple endomorphisms of which are continuous maps satisfying It has recently been discovered that surjective isometries of certain substructures of groups equipped with metrics which are in a way compatible with the group operations have algebraic properties that relate them rather closely to Jordan triple morphisms. This makes us possible to use our structural results to describe all surjective isometries of that correspond to any member of a large class of metrics generalizing the geodesic distance in the natural Riemannian structure on . Finally, we determine the isometry group of relative to a very recently introduced metric that originates from the divergence called Stein’s loss.