Lemmens, Bas,
Roelands, Mark
(2015)
*
Unique geodesics for Thompson's metric.
*
Annales de l’Institut Fourier (Grenoble),
65
(1).
pp. 315-348.
E-ISSN 1777-5310.
(doi:10.5802/aif.2932)
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Official URL https://doi.org/10.5802/aif.2932 |

## Abstract

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique Thompson's metric geodesic connecting \(x\) and \(y\) in the cone of positive self-adjoint elements in a unital \(C^*\)-algebra if, and only if, the spectrum of \(x^{-1/2}yx^{-1/2}\) is contained in \(\{1/\beta,\beta\}\) for some \(\beta ≥ 1\). A similar result will be established for symmetric cones. Secondly, it will be shown that if \(C^°\) is the interior of a finite-dimensional closed cone \(C\), then the Thompson's metric space \((C^°,d_C)\) can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, \(C\) is a polyhedral cone. Moreover, \((C^°,d_C)\) is isometric to a finite-dimensional normed space if, and only if, \(C\) is a simplicial cone. It will also be shown that if \(C^°\) is the interior of a strictly convex cone \(C\) with \(3 ≤ dim \space C ≤ \infty\), then every Thompson's metric isometry is projectively linear.

Item Type: | Article |
---|---|

DOI/Identification number: | 10.5802/aif.2932 |

Additional information: | Open Access |

Subjects: |
Q Science Q Science > QA Mathematics (inc Computing science) Q Science > QA Mathematics (inc Computing science) > QA440 Geometry |

Divisions: | Faculties > Sciences > School of Mathematics Statistics and Actuarial Science |

Depositing User: | Bas Lemmens |

Date Deposited: | 29 May 2014 10:37 UTC |

Last Modified: | 19 Feb 2020 12:04 UTC |

Resource URI: | https://kar.kent.ac.uk/id/eprint/41203 (The current URI for this page, for reference purposes) |

Lemmens, Bas: | https://orcid.org/0000-0001-6713-7683 |

Roelands, Mark: | https://orcid.org/0000-0002-8885-9156 |

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