Unique geodesics for Thompson's metric

الأصل · EN

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⁻¹/²yx⁻¹/² is contained in {1/β,β} for some β≥ 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∘,dC) can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, C is a polyhedral cone. Moreover, (C∘,dC) 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≤ C<∞, then every Thompson's metric isometry is projectively linear.