no. 5
Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces
[Géométries de Hilbert et de Thompson qui sont isométriques à des espaces de Banach en dimension infinie]
Walsh, Cormac1
1 Inria & CMAP, École Polytechnique Université Paris-Saclay 91128 Palaiseau France
Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 1831-1877.

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.

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.

Reçu le :
Révisé le :
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DOI : 10.5802/aif.3198
Classification : 46A40, 46B04, 46A55
Keywords: Hilbert metric, cone, isometry, Banach space, horofunction boundary
Mot clés : Hilbert metric, cone, isometry, Banach space, horofunction boundary

Walsh, Cormac 1

1 Inria & CMAP, École Polytechnique Université Paris-Saclay 91128 Palaiseau France
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Walsh, Cormac. Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces. Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 1831-1877. doi : 10.5802/aif.3198. https://aif.centre-mersenne.org/articles/10.5802/aif.3198/

[1] Akian, Marianne; Gaubert, Stéphane; Walsh, Cormac The max-plus Martin boundary, Doc. Math., Volume 14 (2009), pp. 195-240 | MR | Zbl

[2] Alfsen, Erik M. Compact convex sets and boundary integrals, 57, Springer, 1971, x+210 pages (Ergebnisse der Mathematik und ihrer Grenzgebiete) | MR | Zbl

[3] Aliprantis, Charalambos D.; Border, Kim C. Infinite dimensional analysis: a hitchhiker’s guide, Springer, 2006, xxii+703 pages | MR | Zbl

[4] Aliprantis, Charalambos D.; Burkinshaw, Owen Principles of real analysis, Academic Press, 1998, xii+415 pages | MR | Zbl

[5] Aliprantis, Charalambos D.; Tourky, Rabee Cones and duality, Graduate Studies in Mathematics, 84, American Mathematical Society, 2007, xiv+279 pages | DOI | MR | Zbl

[6] Baranov, Anton; Woracek, Harald Majorization in de Branges spaces. III. Division by Blaschke products, Algebra Anal., Volume 21 (2009) no. 6, pp. 3-46 | DOI | MR | Zbl

[7] Bauer, Heinz Kennzeichnung kompakter Simplexe mit abgeschlossener Extremalpunktmenge, Arch. Math., Volume 14 (1963), pp. 415-421 | DOI | MR | Zbl

[8] Beer, Gerald Topologies on closed and closed convex sets, Mathematics and its Applications, 268, Kluwer Academic Publishers Group, 1993, xii+340 pages | Zbl

[9] Bosché, Aurélien Symmetric cones, the Hilbert and Thompson metrics (2012) (https://arxiv.org/abs/1207.3214)

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

[11] Funk, Paul Über Geometrien, bei denen die Geraden die Kürzesten sind, Math. Ann., Volume 101 (1929) no. 1, pp. 226-237 | DOI | MR | Zbl

[12] Gromov, Mikhael 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 (1981), pp. 183-213 | Zbl

[13] de la Harpe, Pierre 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] Lemmens, Bas; Roelands, Mark; Wortel, Marten Isometries of infinite dimensional Hilbert geometries (https://arxiv.org/abs/1405.4147)

[15] Lemmens, Bas; Roelands, Mark; Wortel, Marten Hilbert and Thompson isometries on cones in JB-algebras (2016) (https://arxiv.org/abs/1609.03473)

[16] Lemmens, Bas; Walsh, Cormac Isometries of polyhedral Hilbert geometries, J. Topol. Anal., Volume 3 (2011) no. 2, pp. 213-241 | DOI | MR | Zbl

[17] Matveev, Vladimir; Troyanov, Marc Isometries of two dimensional Hilbert geometries, Enseign. Math., Volume 61 (2015) no. 3-4, pp. 453-460 | DOI | MR | Zbl

[18] Molnár, Lajos 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] Molnár, Lajos; Nagy, Gergő Thompson isometries on positive operators: the 2-dimensional case, Electron. J. Linear Algebra, Volume 20 (2010), pp. 79-89 | MR | Zbl

[20] Nussbaum, Roger D. Hilbert’s projective metric and iterated nonlinear maps, Memoirs of the American Mathematical Society, 391, American Mathematical Society, 1988, 137 pages | MR | Zbl

[21] Rieffel, Marc A. Group C * -algebras as compact quantum metric spaces, Doc. Math., Volume 7 (2002), pp. 605-651 | MR | Zbl

[22] Speer, Timothy Isometries of the Hilbert Metric, University of California (USA) (2014) (Ph. D. Thesis)

[23] Walsh, Cormac The horofunction boundary of finite-dimensional normed spaces, Math. Proc. Camb. Philos. Soc., Volume 142 (2007) no. 3, pp. 497-507 | Zbl

[24] Walsh, Cormac The horofunction boundary of the Hilbert geometry, Adv. Geom., Volume 8 (2008) no. 4, pp. 503-529 | DOI | MR | Zbl

[25] Walsh, Cormac 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] Walsh, Cormac 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] Walsh, Cormac 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] Walsh, Cormac Gauge-reversing maps on cones, and Hilbert and Thompson isometries, Geom. Topol., Volume 22 (2018) no. 1, pp. 55-104 | Zbl

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