Unique geodesics for Thompson’s metric

Lemmens, Bas; Roelands, Mark

Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 315-348.

Nous présentons une caractérisation géométrique des géodésiques uniques des espaces métriques de Thompson. Nous utilisons cette caractérisation pour démontrer plusieurs autres résultats géométriques. D’abord, nous démontrons qu’il existe une géodésique unique de la métrique de Thompson entre $x$ and $y$ dans le cône d’éléments positifs autoadjoints dans une $C^{*}$-algèbre unitale si et seulement s’il existe $\beta \ge 1$ tel que le spectre de $x^{-1/2}y{x}^{-1/2}$ soit contenu dans $\{1/\beta ,\beta \}$. Un résultat similaire est établi pour des cônes symétriques. Ensuite, nous démontrons que si $C^{\circ }$ est l’intérieur d’un cône fermé $C$ de dimension finie, il existe un plongement quasi-isométrique de l’espace métrique de Thompson $(C^{\circ },d_C)$ dans un espace normé de dimension finie si et seulement si $C$ est un cône polyédrale. De plus, $(C^{\circ },d_C)$ est isométrique à un espace normé de dimension finie si et seulement si $C$ est un cône simplicial. Par ailleurs, il est établi que pour $C^{\circ }$ l’intérieur d’un cône $C$ strictement convexe avec $3\le dimC<\infty$, chaque isométrie de la métrique de Thompson est projectivement linéaire.

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 $x^{-1/2}y{x}^{-1/2}$ is contained in $\{1/\beta ,\beta \}$ for some $\beta \ge 1$. A similar result will be established for symmetric cones. Secondly, it will be shown that if $C^{\circ }$ is the interior of a finite-dimensional closed cone $C$, then the Thompson’s metric space $(C^{\circ },d_C)$ can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, $C$ is a polyhedral cone. Moreover, $(C^{\circ },d_C)$ is isometric to a finite-dimensional normed space if, and only if, $C$ is a simplicial cone. It will also be shown that if $C^{\circ }$ is the interior of a strictly convex cone $C$ with $3\le dimC<\infty$, then every Thompson’s metric isometry is projectively linear.

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