A characterisation of octahedrality in Lipschitz-free spaces

Procházka, Antonín; Rueda Zoca, Abraham

Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 569-588.

On caractérise l’octaédralité de la norme d’un espace Lipschitz libre par le biais d’une nouvelle propriété géométrique de l’espace métrique sous-jacent. Nous étudions les espaces métriques avec et sans cette propriété. Par exemple, les espaces sans cette propriété ne se plongent pas isométriquement dans ${\ell }_1$ et certains espaces de Banach similaires.

We characterise the octahedrality of Lipschitz-free space norm in terms of a new geometric property of the underlying metric space. We study the metric spaces with and without this property. Quite surprisingly, metric spaces without this property cannot be embedded isometrically into ${\ell }_1$ and similar Banach spaces.

Classification : 46B04, 46B20, 46B85
Keywords: Octahedrality, Free spaces, Uniformly discrete metric spaces
Mot clés : Octaédralité, Espaces Lipschitz libres, Espaces métriques uniformément discrets

[1] Abrahamsen, Trond A. Linear extensions, almost isometries, and diameter two, Extr. Math., Volume 30 (2015) no. 2, pp. 135-151 | MR | Zbl

[2] Becerra Guerrero, Julio; López-Pérez, Ginés; Rueda Zoca, Abraham Octahedrality in Lipschitz-free Banach spaces (to appear in Proc. Roy. Soc. Edinburgh Sect. A)

[3] Becerra Guerrero, Julio; López-Pérez, Ginés; Rueda Zoca, Abraham Octahedral norms and convex combination of slices in Banach spaces, J. Funct. Anal., Volume 266 (2014) no. 4, pp. 2424-2435 | DOI | MR | Zbl

[4] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999, xxii+643 pages | DOI | MR | Zbl

[5] Castillo, Jesús M. F.; Papini, P. L.; Simões, Marilda Thick coverings for the unit ball of a Banach space, Houston J. Math., Volume 41 (2015) no. 1, pp. 177-186 | MR | Zbl

[6] Cúth, Marek; Doucha, Michal; Wojtaszczyk, Przemysław On the structure of Lipschitz-free spaces, Proc. Am. Math. Soc., Volume 144 (2016) no. 9, pp. 3833-3846 | DOI | MR | Zbl

[7] Cúth, Marek; Johanis, Michal Isometric embedding of ${\ell }_1$ into Lipschitz-free spaces and ${\ell }_{\infty }$ into their duals, Proc. Am. Math. Soc., Volume 145 (2017) no. 8, pp. 3409-3421 | DOI | MR | Zbl

[8] Cúth, Marek; Kalenda, Ondřej F. K.; Kaplický, Petr Finitely additive measures and complementability of Lipschitz-free spaces (2017) (https://arxiv.org/abs/1703.08384)

[9] Dalet, Aude; Kaufmann, Pedro L.; Procházka, Antonín Characterization of metric spaces whose free space is isometric to ${\ell }_1$, Bull. Belg. Math. Soc. - Simon Stevin, Volume 23 (2016) no. 3, pp. 391-400 http://projecteuclid.org/euclid.bbms/1473186513 | MR | Zbl

[10] Fabian, Marián; Habala, Petr; Hájek, Petr; Montesinos, Vicente; Zizler, Václav Banach space theory. The basis for linear and nonlinear analysis, CMS Books in Mathematics, Springer, 2011, xiv+820 pages | DOI | MR | Zbl

[11] García Lirola, Luis Carlos Convexity, Optimization and Geometry of the Ball in Banach Spaces, Universidad de Murcia (Spain) (2017) (Ph. D. Thesis)

[12] Ghoussoub, Nassif; Godefroy, Gilles; Maurey, Bernard; Schachermayer, Walter Some topological and geometrical structures in Banach spaces, Mem. Am. Math. Soc., Volume 70 (1987) no. 378, iv+116 pages | DOI | MR | Zbl

[13] Godard, Alexandre Tree metrics and their Lipschitz-free spaces, Proc. Am. Math. Soc., Volume 138 (2010) no. 12, pp. 4311-4320 | DOI | MR | Zbl

[14] Godefroy, Gilles Metric characterization of first Baire class linear forms and octahedral norms, Stud. Math., Volume 95 (1989) no. 1, pp. 1-15 | DOI | MR | Zbl

[15] Godefroy, Gilles A survey on Lipschitz-free Banach spaces, Commentat. Math., Volume 55 (2015) no. 2, pp. 89-118 | MR | Zbl

[16] Godefroy, Gilles; Kalton, Nigel John The ball topology and its applications, Banach space theory (Iowa City, IA, 1987) (Contemporary Mathematics), Volume 85, American Mathematical Society, 1989, pp. 195-237 | DOI | MR | Zbl

[17] Godefroy, Gilles; Kalton, Nigel John Lipschitz-free Banach spaces, Stud. Math., Volume 159 (2003) no. 1, pp. 121-141 | DOI | MR | Zbl

[18] Haller, Rainis; Langemets, Johann; Põldvere, Märt On duality of diameter 2 properties, J. Convex Anal., Volume 22 (2015) no. 2, pp. 465-483 | MR | Zbl

[19] Ivakhno, Yevgen; Kadets, Vladimir; Werner, Dirk The Daugavet property for spaces of Lipschitz functions, Math. Scand., Volume 101 (2007) no. 2, pp. 261-279 corrigendum ibid. 104 (2009), no. 2, p. 319 | DOI | MR | Zbl

[20] Johnson, William B.; Lindenstrauss, Joram; Preiss, David; Schechtman, Gideon Almost Fréchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. Lond. Math. Soc., Volume 84 (2002) no. 3, pp. 711-746 | DOI | MR | Zbl

[21] Johnson, William B.; Lindenstrauss, Joram; Preiss, David; Schechtman, Gideon Lipschitz quotients from metric trees and from Banach spaces containing $l_1$, J. Funct. Anal., Volume 194 (2002) no. 2, pp. 332-346 | DOI | MR | Zbl

[22] Maurey, Bernard Types and $l_1$-subspaces, Texas functional analysis seminar 1982–1983 (Austin, Tex.) (Longhorn Notes), University of Texas, 1983, pp. 123-137 | MR | Zbl

[23] McShane, Edward James Extension of range of functions, Bull. Am. Math. Soc., Volume 40 (1934) no. 12, pp. 837-842 | DOI | MR | Zbl

[24] Milman, Vitali D. Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspehi Mat. Nauk, Volume 26 (1971) no. 6(162), pp. 73-149 | MR | Zbl

[25] Petitjean, Colin Lipschitz-free spaces and Schur properties, J. Math. Anal. Appl., Volume 453 (2017) no. 2, pp. 894-907 | DOI | Zbl

[26] Yagoub-Zidi, Yamina Some isometric properties of subspaces of function spaces, Mediterr. J. Math., Volume 10 (2013) no. 4, pp. 1905-1915 | DOI | MR | Zbl