We study the horofunction boundaries of Hilbert and Thompson geometries, and of Banach spaces, in arbitrary dimension. By comparing the boundaries of these spaces, we show that the only Hilbert and Thompson geometries that are isometric to Banach spaces are the ones defined on the cone of positive continuous functions on a compact space.
Nous étudions les frontières d’horofonctions des géométries de Hilbert et de Thompson, et des espaces de Banach, en dimension arbitraire. En comparant les frontières de ces espaces, nous montrons que les seules géométries de Hilbert et de Thompson qui sont isométriques à des espaces de Banach sont celles qui sont définies sur le cône de fonctions continues positives sur un espace compact.
Revised:
Accepted:
Published online:
Keywords: Hilbert metric, cone, isometry, Banach space, horofunction boundary
Mot clés : Hilbert metric, cone, isometry, Banach space, horofunction boundary
@article{AIF_2018__68_5_1831_0, author = {Walsh, Cormac}, title = {Hilbert and {Thompson} geometries isometric to infinite-dimensional {Banach} spaces}, journal = {Annales de l'Institut Fourier}, pages = {1831--1877}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {5}, year = {2018}, doi = {10.5802/aif.3198}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.3198/} }
TY - JOUR AU - Walsh, Cormac TI - Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces JO - Annales de l'Institut Fourier PY - 2018 SP - 1831 EP - 1877 VL - 68 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3198/ DO - 10.5802/aif.3198 LA - en ID - AIF_2018__68_5_1831_0 ER -
%0 Journal Article %A Walsh, Cormac %T Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces %J Annales de l'Institut Fourier %D 2018 %P 1831-1877 %V 68 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3198/ %R 10.5802/aif.3198 %G en %F AIF_2018__68_5_1831_0
Walsh, Cormac. Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces. Annales de l'Institut Fourier, Volume 68 (2018) no. 5, pp. 1831-1877. doi : 10.5802/aif.3198. http://archive.numdam.org/articles/10.5802/aif.3198/
[1] The max-plus Martin boundary, Doc. Math., Volume 14 (2009), pp. 195-240 | MR | Zbl
[2] Compact convex sets and boundary integrals, 57, Springer, 1971, x+210 pages (Ergebnisse der Mathematik und ihrer Grenzgebiete) | MR | Zbl
[3] Infinite dimensional analysis: a hitchhiker’s guide, Springer, 2006, xxii+703 pages | MR | Zbl
[4] Principles of real analysis, Academic Press, 1998, xii+415 pages | MR | Zbl
[5] Cones and duality, Graduate Studies in Mathematics, 84, American Mathematical Society, 2007, xiv+279 pages | DOI | MR | Zbl
[6] Majorization in de Branges spaces. III. Division by Blaschke products, Algebra Anal., Volume 21 (2009) no. 6, pp. 3-46 | DOI | MR | Zbl
[7] Kennzeichnung kompakter Simplexe mit abgeschlossener Extremalpunktmenge, Arch. Math., Volume 14 (1963), pp. 415-421 | DOI | MR | Zbl
[8] Topologies on closed and closed convex sets, Mathematics and its Applications, 268, Kluwer Academic Publishers Group, 1993, xii+340 pages | Zbl
[9] Symmetric cones, the Hilbert and Thompson metrics (2012) (https://arxiv.org/abs/1207.3214)
[10] Hilbert metrics and Minkowski norms, J. Geom., Volume 83 (2005) no. 1-2, pp. 22-31 | DOI | MR | Zbl
[11] Über Geometrien, bei denen die Geraden die Kürzesten sind, Math. Ann., Volume 101 (1929) no. 1, pp. 226-237 | DOI | MR | Zbl
[12] Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) (Annals of Mathematics Studies), Volume 97, Princeton University Press (1981), pp. 183-213 | Zbl
[13] On Hilbert’s metric for simplices, Geometric group theory, Vol. 1 (Sussex, 1991) (London Mathematical Society Lecture Note Series), Volume 181, Cambridge University Press, 1993, pp. 97-119 | MR | Zbl
[14] Isometries of infinite dimensional Hilbert geometries (https://arxiv.org/abs/1405.4147)
[15] Hilbert and Thompson isometries on cones in JB-algebras (2016) (https://arxiv.org/abs/1609.03473)
[16] Isometries of polyhedral Hilbert geometries, J. Topol. Anal., Volume 3 (2011) no. 2, pp. 213-241 | DOI | MR | Zbl
[17] Isometries of two dimensional Hilbert geometries, Enseign. Math., Volume 61 (2015) no. 3-4, pp. 453-460 | DOI | MR | Zbl
[18] Thompson isometries of the space of invertible positive operators, Proc. Am. Math. Soc., Volume 137 (2009) no. 11, pp. 3849-3859 | DOI | MR | Zbl
[19] Thompson isometries on positive operators: the 2-dimensional case, Electron. J. Linear Algebra, Volume 20 (2010), pp. 79-89 | MR | Zbl
[20] Hilbert’s projective metric and iterated nonlinear maps, Memoirs of the American Mathematical Society, 391, American Mathematical Society, 1988, 137 pages | MR | Zbl
[21] Group -algebras as compact quantum metric spaces, Doc. Math., Volume 7 (2002), pp. 605-651 | MR | Zbl
[22] Isometries of the Hilbert Metric, University of California (USA) (2014) (Ph. D. Thesis https://arxiv.org/abs/1411.1826)
[23] The horofunction boundary of finite-dimensional normed spaces, Math. Proc. Camb. Philos. Soc., Volume 142 (2007) no. 3, pp. 497-507 | Zbl
[24] The horofunction boundary of the Hilbert geometry, Adv. Geom., Volume 8 (2008) no. 4, pp. 503-529 | DOI | MR | Zbl
[25] Minimum representing measures in idempotent analysis, Tropical and idempotent mathematics (Contemporary Mathematics), Volume 495, American Mathematical Society, 2009, pp. 367-382 | DOI | MR | Zbl
[26] The horoboundary and isometry group of Thurston’s Lipschitz metric, Handbook of Hilbert geometry. Volume IV (IRMA Lectures in Mathematics and Theoretical Physics), Volume 19, European Mathematical Society, 2014, pp. 327-353 | DOI | MR | Zbl
[27] The horofunction boundary and isometry group of the Hilbert geometry, Handbook of Hilbert geometry (IRMA Lectures in Mathematics and Theoretical Physics), Volume 22, European Mathematical Society, 2014, pp. 127-146 | MR
[28] Gauge-reversing maps on cones, and Hilbert and Thompson isometries, Geom. Topol., Volume 22 (2018) no. 1, pp. 55-104 | Zbl
Cited by Sources: