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Unique geodesics for Thompson's metric

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) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided. (Contact us about this Publication)
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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