### Abstract

In this paper, we determine the structure of certain algebraic morphisms and isometries of the space (Formula presented.) of all (Formula presented.) complex positive definite matrices. In the case (Formula presented.) , we describe all continuous Jordan triple endomorphisms of (Formula presented.) which are continuous maps (Formula presented.) satisfying (Formula presented.) 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 (Formula presented.) that correspond to any member of a large class of metrics generalizing the geodesic distance in the natural Riemannian structure on (Formula presented.). Finally, we determine the isometry group of (Formula presented.) relative to a very recently introduced metric that originates from the divergence called Stein’s loss.

Original language | English |
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Pages (from-to) | 12-33 |

Number of pages | 22 |

Journal | Linear and Multilinear Algebra |

Volume | 63 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 7 2015 |

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### Keywords

- Jordan triple endomorphisms
- geodesic distance
- isometries
- positive definite matrices
- symmetric Stein divergence
- unitarily invariant norms

### ASJC Scopus subject areas

- Algebra and Number Theory