A characterisation of the Daugavet property in spaces of Lipschitz functions

Transcription

1 A characterisation of the Daugavet property in spaces of Lipschitz functions Luis C. García-Lirola Joint work with Antonin Procházka and Abraham Rueda Zoca Universidad de Murcia Septièmes journées Besançon-Neuchâtel d analyse fonctionnelle June, 2017 Research partially supported by

2 The Daugavet property Definition A Banach space X is said to have the Daugavet property if I + T = 1 + T for every rank-one operator T : X X. L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

3 The Daugavet property Definition A Banach space X is said to have the Daugavet property if I + T = 1 + T for every rank-one operator T : X X. Examples of spaces with the Daugavet property: C[0, 1] (Daugavet, 1963), L 1 [0, 1] (Lozanovskii, 1966), L [0, 1] (Pelczynski, 1965). X has the Daugavet property whenever X has the Daugavet property. L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

4 The Daugavet property Definition A Banach space X is said to have the Daugavet property if I + T = 1 + T for every rank-one operator T : X X. Examples of spaces with the Daugavet property: C[0, 1] (Daugavet, 1963), L 1 [0, 1] (Lozanovskii, 1966), L [0, 1] (Pelczynski, 1965). X has the Daugavet property whenever X has the Daugavet property. Theorem (Kadets-Shvidkoy-Sirotkin-Werner, 2000) X has the Daugavet property if and only if for every x 0 S X, every ε > 0 and every slice S of B X there exists x S such that x 0 + x > 2 ε. L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

5 Spaces of Lipschitz functions Given a metric space (M, d) and a distinguished point 0 M, the space Lip 0 (M) := {f : M R : f is Lipschitz, f (0) = 0} is a dual Banach space when equipped with the norm { } f (x) f (y) f L := sup : x y. d(x, y) The canonical predual of Lip 0 (M) is the Lipschitz-free space F(M) = span{δ x : x M}. We will assume that M is complete. L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

6 Daugavet property for spaces of Lipschitz functions Note that Lip 0 ([0, 1]) is isometric to L [0, 1] and thus it has the Daugavet property. Does Lip 0 ([0, 1] 2 ) have the Daugavet property? (D. Werner, 2001) L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

7 Daugavet property for spaces of Lipschitz functions Note that Lip 0 ([0, 1]) is isometric to L [0, 1] and thus it has the Daugavet property. Does Lip 0 ([0, 1] 2 ) have the Daugavet property? (D. Werner, 2001) This problem was solved by Ivakhno, Kadets and Werner in L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

8 A metric space M is said to be a length space if for every x, y M, the distance d(x, y) is equal to the infimum of the length of rectifiable curves joining them. Moreover, if that infimum is always attained then we will say that M is a geodesic space. local if for every ε > 0 and every f Lip 0 (M) there exist u, v M f (u) f (v) such that 0 < d(u, v) < ε and d(u,v) > f L ε. spreadingly local if for every ε > 0 and every f Lip 0 (M) the set { } x M : inf f L B(x,δ) > f L ε δ>0 is infinite. (Z) if for every ε > 0 and every x, y M, x y, there is z M \ {x, y} such that d(x, z) + d(z, y) d(x, y) + ε min{d(x, z), d(z, y)} Theorem (Ivakhno Kadets Werner, 2006) See diagram. L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

9 Main result Theorem (G.-L. Procházka Rueda-Zoca) Let M be a complete metric space. The following are equivalent: (i) M is a length space. (ii) Lip 0 (M) has the Daugavet property. (iii) F(M) has the Daugavet property. Remark that a complete metric M is a length space if, and only if, the following condition hold: x, y M δ > 0 z M : max{d(x, z), d(y, z)} 1 + δ d(x, y) 2 L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

10 Assume that F(M) has the Daugavet property. Then M is local. L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

11 Assume that F(M) has the Daugavet property. Then M is local. Step 1. For every f S Lip0 (M), ε > 0 and x, y M there are u, v M such that f (u) f (v) d(u,v) > 1 ε and (1 ε)(d(x, y) + d(u, v)) < min{d(x, v) + d(u, y), d(x, u) + d(v, y)} Sketch of the proof: The geometric characterisation of the Daugavet property provides u, v M such that δu δv d(u,v) S(B F(M), f, ε) and δ x δ y d(x, y) + δ u δ v d(u, v) > 2 ε L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

12 Assume that F(M) has the Daugavet property. Then M is local. Step 1. For every f S Lip0 (M), ε > 0 and x, y M there are u, v M such that f (u) f (v) d(u,v) > 1 ε and (1 ε)(d(x, y) + d(u, v)) < min{d(x, v) + d(u, y), d(x, u) + d(v, y)} Step 2. For every f S Lip0 (M), ε > 0 and x, y M such that f (x) f (y) d(x,y) > 1 ε there are u, v M such that d(u, v) < ε d(x, y). (1 ε) 2 f (u) f (v) d(u,v) > 1 ε and Sketch of the proof: Apply Step 1 to the function g = i=1 f i where f 1 = f, f 2 (t) = d(y, t), f 3 (t) = d(x, t) and f 4 (t) = d(x, y) d(t, y) d(t, x) 2 d(t, y) + d(t, x) L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

13 Assume that F(M) has the Daugavet property. Then M is local. Step 1. For every f S Lip0 (M), ε > 0 and x, y M there are u, v M such that f (u) f (v) d(u,v) > 1 ε and (1 ε)(d(x, y) + d(u, v)) < min{d(x, v) + d(u, y), d(x, u) + d(v, y)} Step 2. For every f S Lip0 (M), ε > 0 and x, y M such that f (x) f (y) d(x,y) > 1 ε there are u, v M such that d(u, v) < ε d(x, y). (1 ε) 2 f (u) f (v) d(u,v) > 1 ε and Finally, given f S Lip0 (M) use Step 2 to find sequences (x n ), (y n ) with and f (xn) f (yn) d(x n,y n) > 1 ε. ( ) ε n d(x n, y n ) < (1 ε) 2 d(x 0, y 0 ) L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

14 Strongly exposed points of B F(M) Theorem (G.-L. Procházka Rueda-Zoca) Let x, y M, x y. The following are equivalent. δx δy (i) d(x,y) is a strongly exposed point of B F(M). (ii) There is f Lip 0 (M) peaking at (x, y), that is, lim n f (u n ) f (v n ) d(u n, v n ) = 1 lim n u n = x, lim n v n = y (iii) There is ε > 0 such that for every z M \ {x, y}, d(x, z) + d(y, z) > d(x, y) + ε min{d(x, z), d(y, z)} Corollary (G.-L. Procházka Rueda-Zoca) Let M be a compact metric space. Then Lip 0 (M) has the Daugavet property if and only if B F(M) does not have any strongly exposed point. L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

15 References Dalet, A., P. Kaufmann, and A. Procházka. Characterization of metric spaces whose free space is isometric to l 1. In: Bull. Belg. Math. Soc. Simon Stevin 23.3 (2016), pp García-Lirola, L., A. Procházka, and A. Rueda Zoca. A characterisation of the Daugavet property in spaces of Lipschitz functions. arxiv: Godard, A. Tree metrics and their Lipschitz-free spaces. In: Proc. Amer. Math. Soc (2010), pp Ivakhno, Y., V. Kadets, and D. Werner. The Daugavet property for spaces of Lipschitz functions. In: Math. Scand (2007), pp Kadets, V. et al. Banach spaces with the Daugavet property. In: Trans. Amer. Math. Soc (2000), pp Procházka, A. and A. Rueda Zoca. A characterisation of octahedrality in Lipschitz-free spaces. arxiv: Werner, D. Recent progress on the Daugavet property. In: Irish Math. Soc. Bull. 46 (2001), pp L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9

16 References Dalet, A., P. Kaufmann, and A. Procházka. Characterization of metric spaces whose free space is isometric to l 1. In: Bull. Belg. Math. Soc. Simon Stevin 23.3 (2016), pp García-Lirola, L., A. Procházka, and A. Rueda Zoca. A characterisation of the Daugavet property in spaces of Lipschitz functions. arxiv: Godard, A. Tree metrics and their Lipschitz-free spaces. In: Proc. Amer. Math. Soc (2010), pp Ivakhno, Y., V. Kadets, and D. Werner. The Daugavet property for spaces of Lipschitz functions. In: Math. Scand (2007), pp Kadets, V. et al. Banach spaces with the Daugavet property. In: Trans. Amer. Math. Soc (2000), pp Procházka, A. and A. Rueda Zoca. A characterisation of octahedrality in Lipschitz-free spaces. arxiv: Werner, D. Recent progress on the Daugavet property. In: Irish Math. Soc. Bull. 46 (2001), pp Thank you for your attention L. Garcia-Lirola (Universidad de Murcia) A characterisation of the Daugavet property... July, / 9