Isometric embedding of ℓ₁ into Lipschitz-free spaces and ℓ_{∞} into their duals

Cúth, Marek; Johanis, Michal

doi:10.1090/proc/13590

Isometric embedding of $\ell _1$ into Lipschitz-free spaces and $\ell _\infty$ into their duals
HTML articles powered by AMS MathViewer

by Marek Cúth and Michal Johanis PDF

Proc. Amer. Math. Soc. 145 (2017), 3409-3421 Request permission

Abstract:

We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell _\infty$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell _1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. In the last section we survey the relations between isometric embeddability of $\ell _\infty$ into $X^*$ and containment of a good copy of $\ell _1$ in $X$ for a general Banach space $X$.

References

Marek Cúth and Michal Doucha, Lipschitz-free spaces over ultrametric spaces, Mediterr. J. Math. 13 (2016), no. 4, 1893–1906. MR 3530906, DOI 10.1007/s00009-015-0566-7
Marek Cúth, Michal Doucha, and Przemysław Wojtaszczyk, On the structure of Lipschitz-free spaces, Proc. Amer. Math. Soc. 144 (2016), no. 9, 3833–3846. MR 3513542, DOI 10.1090/proc/13019
Stephen J. Dilworth, Maria Girardi, and James Hagler, Dual Banach spaces which contain an isometric copy of $L_1$, Bull. Polish Acad. Sci. Math. 48 (2000), no. 1, 1–12. MR 1751149
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
Patrick N. Dowling, Isometric copies of $c_0$ and $l^\infty$ in duals of Banach spaces, J. Math. Anal. Appl. 244 (2000), no. 1, 223–227. MR 1746799, DOI 10.1006/jmaa.2000.6714
P. N. Dowling, C. J. Lennard, and B. Turett, Renormings of $l^1$ and $c_0$ and fixed point properties, Handbook of metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 269–297. MR 1904280
Aude Dalet, Pedro L. Kaufmann, and Antonín Procházka, Characterization of metric spaces whose free space is isometric to $\ell _1$, Bull. Belg. Math. Soc. Simon Stevin 23 (2016), no. 3, 391–400. MR 3545460
Patrick N. Dowling, Narcisse Randrianantoanina, and Barry Turett, Remarks on James’s distortion theorems. II, Bull. Austral. Math. Soc. 59 (1999), no. 3, 515–522. MR 1698052, DOI 10.1017/S0004972700033219
Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, and Václav Zizler, Banach space theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. The basis for linear and nonlinear analysis. MR 2766381, DOI 10.1007/978-1-4419-7515-7
M. Fabián, L. Zajíček, and V. Zizler, On residuality of the set of rotund norms on a Banach space, Math. Ann. 258 (1981/82), no. 3, 349–351. MR 649205, DOI 10.1007/BF01450688
G. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003), no. 1, 121–141. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday. MR 2030906, DOI 10.4064/sm159-1-6
Petr Hájek, Vicente Montesinos Santalucía, Jon Vanderwerff, and Václav Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 26, Springer, New York, 2008. MR 2359536
William B. Johnson and Narcisse Randrianantoanina, On complemented versions of James’s distortion theorems, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2751–2757. MR 2317949, DOI 10.1090/S0002-9939-07-08775-8
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056