Unique geodesics for Thompson’s metric
[Les géodésiques uniques de la métrique de Thompson]
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 β1 tel que le spectre de x -1/2 yx -1/2 soit contenu dans {1/β,β}. Un résultat similaire est établi pour des cônes symétriques. Ensuite, nous démontrons que si C est l’intérieur d’un cône fermé C de dimension finie, il existe un plongement quasi-isométrique de l’espace métrique de Thompson (C ,d C ) dans un espace normé de dimension finie si et seulement si C est un cône polyédrale. De plus, (C ,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 l’intérieur d’un cône C strictement convexe avec 3dimC<, 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 yx -1/2 is contained in {1/β,β} for some β1. 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 3dimC<, then every Thompson’s metric isometry is projectively linear.

DOI : 10.5802/aif.2932
Classification : 53C22, 51Fxx, 53C60
Keywords: Geodesics, Thompson’s (part) metric, Hilbert’s (projective) metric, cones, isometries
Mot clés : géodésiques, métrique de Thompson, métrique d’Hilbert, cônes, isométries
Lemmens, Bas 1 ; Roelands, Mark 2

1 School of Mathematics, Statistics & Actuarial Science, Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF, UK.
2 School of Mathematics, Statistics & Actuarial Science, Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF, UK
@article{AIF_2015__65_1_315_0,
     author = {Lemmens, Bas and Roelands, Mark},
     title = {Unique geodesics for {Thompson{\textquoteright}s} metric},
     journal = {Annales de l'Institut Fourier},
     pages = {315--348},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {1},
     year = {2015},
     doi = {10.5802/aif.2932},
     zbl = {06496541},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2932/}
}
TY  - JOUR
AU  - Lemmens, Bas
AU  - Roelands, Mark
TI  - Unique geodesics for Thompson’s metric
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 315
EP  - 348
VL  - 65
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2932/
DO  - 10.5802/aif.2932
LA  - en
ID  - AIF_2015__65_1_315_0
ER  - 
%0 Journal Article
%A Lemmens, Bas
%A Roelands, Mark
%T Unique geodesics for Thompson’s metric
%J Annales de l'Institut Fourier
%D 2015
%P 315-348
%V 65
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2932/
%R 10.5802/aif.2932
%G en
%F AIF_2015__65_1_315_0
Lemmens, Bas; Roelands, Mark. Unique geodesics for Thompson’s metric. Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 315-348. doi : 10.5802/aif.2932. https://aif.centre-mersenne.org/articles/10.5802/aif.2932/

[1] Akian, M.; Gaubert, S.; Lemmens, B.; Nussbaum, R. D. Iteration of order preserving subhomogeneous maps on a cone, Math. Proc. Cambridge Philos. Soc., Volume 140 (2006) no. 1, pp. 157-176 | DOI | MR | Zbl

[2] Andruchow, E.; Corach, G.; Stojanoff, D. Geometrical significance of Löwner-Heinz inequality, Proc. Amer. Math. Soc., Volume 128 (2000) no. 4, pp. 1031-1037 | DOI | MR | Zbl

[3] Bernig, A. Hilbert geometry of polytopes, Arch. Math. (Basel), Volume 92 (2009) no. 4, pp. 314-324 | DOI | MR | Zbl

[4] Birkhoff, G. Extensions of Jentzsch’s theorem, Trans. Amer. Math. Soc., Volume 85 (1957) no. 1, pp. 219-227 | MR | Zbl

[5] Bosché, A. Symmetric cones, the Hilbert and Thompson metrics (arXiv:1207.3214)

[6] Colbois, B.; Verovic, P. Hilbert domains that admit a quasi-isometric embedding into Euclidean space, Adv. Geom., Volume 11 (2011) no. 2, pp. 465-470 | MR | Zbl

[7] Conway, J. B. A course in functional analysis, Graduate Texts in Mathematics, 96, Springer-Verlag, New York, 1990 | MR | Zbl

[8] Corach, G.; Maestripieri, A. L. Differential and metrical structure of positive operators, Positivity, Volume 3 (1999) no. 4, pp. 297-315 | DOI | MR | Zbl

[9] Corach, G.; Porta, H.; Recht, L. Convexity of the geodesic distance on spaces of positive operators, Illinois J. Math., Volume 38 (1994) no. 1, pp. 87-94 | MR | Zbl

[10] de la Harpe, P. On Hilbert’s metric for simplices, Geometric group theory, Vol. 1 (Sussex, 1991) (London Math. Soc. Lecture Note Ser.), Volume 181, Cambridge Univ. Press, Cambridge, 1993, pp. 97-119 | MR | Zbl

[11] Faraut, J.; Korányi, A. Analysis on Symmetric Cones, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1994 | MR | Zbl

[12] Foertsch, T.; Karlsson, A. Hilbert metrics and Minkowski norms, J. Geom., Volume 83 (2005) no. 1-2, pp. 22-31 | DOI | MR | Zbl

[13] Hyers, D. H.; Isac, G.; Rassias, T. M. Topics in nonlinear analysis & applications, World Scientific Publishing Co., Inc., River Edge, NJ,, 1997 | MR | Zbl

[14] Karlsson, A.; Noskov, G. A. The Hilbert metric and Gromov hyperbolicity, Enseign. Math. (2), Volume 48 (2002) no. 1-2, pp. 73-89 | MR | Zbl

[15] Koecher, M. Positivitätsbereiche im n , Amer. J. Math., Volume 79 (1957), pp. 575-596 | DOI | MR | Zbl

[16] Lawson, J.; Lim, Y. Metric convexity of symmetric cones, Osaka J. Math., Volume 44 (2007) no. 4, pp. 795-816 | MR | Zbl

[17] Lemmens, B.; Nussbaum, R. Nonlinear Perron-Frobebius theory, Cambridge Tracts in Mathematics, 189, Cambridge Univ. Press, Cambridge, 2012 | MR | Zbl

[18] Lim, Y. Finsler metrics on symmetric cones, Math. Ann., Volume 316 (2000), pp. 379-389 | DOI | MR | Zbl

[19] Lim, Y. Hilbert’s projective metric on Lorentz cones and Birkhoff formula for Lorentzian compressions, Linear Algebra Appl., Volume 423 (2007) no. 2–3, pp. 246-254 | DOI | MR | Zbl

[20] Lim, Y. Geometry of midpoint sets for Thompson’s metric, Linear Algebra Appl., Volume 439 (2013) no. 1, pp. 211-227 | DOI | MR | Zbl

[21] Lim, Y.; Pálfia, M. Matrix power means and the Karcher mean, J. Funct. Anal., Volume 262 (2012) no. 4, pp. 1498-1514 | DOI | MR | Zbl

[22] Liverani, C.; Wojtkowski, M. P. Generalization of the Hilbert metric to the space of positive definite matrices, Pacific J. of Math., Volume 166 (1994), pp. 339-355 | DOI | MR | Zbl

[23] Molnár, L. Thompson isometries of the space of invertible positive operators, Proc. Amer. Math. Soc., Volume 137 (2009), pp. 3849-3859 | DOI | MR | Zbl

[24] Noll, W.; Schäffer, J. J. Orders, gauge, and distance in faceless linear cones; with examples relevant to continuum mechanics and relativity, Arch. Rational Mech. Anal., Volume 66 (1977) no. 4, pp. 345-377 | DOI | MR | Zbl

[25] Noll, W.; Schäffer, J. J. Order-isomorphisms in affine spaces, Ann. Mat. Pura Appl. (4), Volume 117 (1978), pp. 243-262 | DOI | MR | Zbl

[26] Nussbaum, R. D. Hilbert’s projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc., Volume 391 (1988), pp. 1-137 | MR | Zbl

[27] Nussbaum, R. D. Finsler structures for the part metric and Hilbert’s projective metric and applications to ordinary differential equations, Differential Integral Equations, Volume 7 (1994) no. 5–6, pp. 1649-1707 | MR | Zbl

[28] Nussbaum, R. D.; Walsh, C. A metric inequality for the Thompson and Hilbert geometries, J. Inequal. Pure Appl. Math., Volume 5 (2004) no. 3 (Article 54, 14 pp) | MR | Zbl

[29] Papadopoulos, A. Metric Spaces, Convexity, and Nonpositive Curvature, IRMA Lectures in Mathematics and Theoretical Physics 6, European Math. Soc. Zürich, 2005 | MR | Zbl

[30] Rockafellar, R. T. Convex Analysis, Princeton Landmarks in Mathematics, Princeton, N.J., 1997 | MR | Zbl

[31] Thompson, A. C. On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. Soc., Volume 14 (1963), pp. 438-443 | MR | Zbl

[32] Vinberg, E. B. Homogeneous cones, Soviet Math. Dokl., Volume 1 (1960), pp. 787-790 | MR | Zbl

Cité par Sources :