Unique geodesics for Thompson's metric

TL;DR

This paper characterizes unique geodesics in Thompson's metric spaces, linking geometric properties to spectral conditions and cone structures, and explores embeddings and isometries in finite-dimensional cones.

Contribution

It provides a geometric characterization of unique geodesics in Thompson's metric and establishes conditions for embeddings and isometries related to cone structures.

Findings

Unique geodesics exist iff the spectrum of a certain operator is contained in two points.

Thompson's metric space can be embedded into finite-dimensional normed spaces iff the cone is polyhedral.

All isometries in strictly convex cones are projectively linear.

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 $\beta\geq 1$. 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 $3\leq \dim C<\infty$, then every Thompson's metric isometry is projectively linear.