### 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 |
---|---|

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

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Jordan triple endomorphisms and isometries of spaces of positive definite matrices.** / Molnár, L.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Jordan triple endomorphisms and isometries of spaces of positive definite matrices

AU - Molnár, L.

PY - 2015/1/7

Y1 - 2015/1/7

N2 - 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.

AB - 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.

KW - geodesic distance

KW - isometries

KW - Jordan triple endomorphisms

KW - positive definite matrices

KW - symmetric Stein divergence

KW - unitarily invariant norms

UR - http://www.scopus.com/inward/record.url?scp=84908548000&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908548000&partnerID=8YFLogxK

U2 - 10.1080/03081087.2013.844231

DO - 10.1080/03081087.2013.844231

M3 - Article

AN - SCOPUS:84908548000

VL - 63

SP - 12

EP - 33

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

IS - 1

ER -