On the structure of Lipschitz-free spaces

Cúth, Marek; Doucha, Michal; Wojtaszczyk, Przemysław

doi:10.1090/proc/13019

On the structure of Lipschitz-free spaces
HTML articles powered by AMS MathViewer

by Marek Cúth, Michal Doucha and Przemysław Wojtaszczyk PDF

Proc. Amer. Math. Soc. 144 (2016), 3833-3846 Request permission

Abstract:

In this note we study the structure of Lipschitz-free Banach spaces. We show that every Lipschitz-free Banach space over an infinite metric space contains a complemented copy of $\ell _1$. This result has many consequences for the structure of Lipschitz-free Banach spaces. Moreover, we give an example of a countable compact metric space $K$ such that $\mathcal {F}(K)$ is not isomorphic to a subspace of $L_1$ and we show that whenever $M$ is a subset of $\mathbb {R}^n$, then $\mathcal {F}(M)$ is weakly sequentially complete; in particular, $c_0$ does not embed into $\mathcal {F}(M)$.

References

Israel Aharoni, Every separable metric space is Lipschitz equivalent to a subset of $c^{+}_{0}$, Israel J. Math. 19 (1974), 284–291. MR 511661, DOI 10.1007/BF02757727
Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
J. Bourgain, On weak completeness of the dual of spaces of analytic and smooth functions, Bull. Soc. Math. Belg. Sér. B 35 (1983), no. 1, 111–118. MR 712065
J. Bourgain, The Dunford-Pettis property for the ball-algebras, the polydisc-algebras and the Sobolev spaces, Studia Math. 77 (1984), no. 3, 245–253. MR 745280, DOI 10.4064/sm-77-3-246-253
Marek Cúth and Michal Doucha, Lipschitz-free spaces over ultrametric spaces, Mediterr. J. Math., electronically published on May 03, 2015, DOI: http://dx.doi.org/10.1007/s00009-015-0566-7 (to appear in print).
A. Dalet, Free spaces over countable compact metric spaces, Proc. Amer. Math. Soc. 143 (2015), no. 8, 3537–3546. MR 3348795, DOI 10.1090/S0002-9939-2015-12518-X
Yves Dutrieux and Valentin Ferenczi, The Lipschitz free Banach spaces of $C(K)$-spaces, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1039–1044. MR 2196036, DOI 10.1090/S0002-9939-05-08301-2
Charles Fefferman, Interpolation and extrapolation of smooth functions by linear operators, Rev. Mat. Iberoamericana 21 (2005), no. 1, 313–348. MR 2155023, DOI 10.4171/RMI/424
V. P. Fonf and P. Wojtaszczyk, Properties of the Holmes space, Topology Appl. 155 (2008), no. 14, 1627–1633. MR 2435154, DOI 10.1016/j.topol.2008.03.008
Joanna Garbulińska and Wiesław Kubiś, Remarks on Gurariĭ spaces, Extracta Math. 26 (2011), no. 2, 235–269. MR 2977626
A. Godard, Tree metrics and their Lipschitz-free spaces, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4311–4320. MR 2680057, DOI 10.1090/S0002-9939-2010-10421-5
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
V. I. Gurariĭ, Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Sibirsk. Mat. Ž. 7 (1966), 1002–1013 (Russian). MR 0200697
Petr Hájek and Michal Johanis, Smooth analysis in Banach spaces, De Gruyter Series in Nonlinear Analysis and Applications, vol. 19, De Gruyter, Berlin, 2014. MR 3244144, DOI 10.1515/9783110258998
Pedro Levit Kaufmann, Products of Lipschitz-free spaces and applications, Studia Math. 226 (2015), no. 3, 213–227. MR 3356002, DOI 10.4064/sm226-3-2
Pedro Levit Kaufmann, Products of Lipschitz-free spaces and applications, Studia Math. 226 (2015), no. 3, 213–227. MR 3356002, DOI 10.4064/sm226-3-2
S. V. Kisljakov, Sobolev imbedding operators, and the nonisomorphism of certain Banach spaces, Funkcional. Anal. i Priložen. 9 (1975), no. 4, 22–27 (Russian). MR 0627173
G. Lancien and E. Pernecká, Approximation properties and Schauder decompositions in Lipschitz-free spaces, J. Funct. Anal. 264 (2013), no. 10, 2323–2334. MR 3035057, DOI 10.1016/j.jfa.2013.02.012
J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 231188, DOI 10.4064/sm-29-3-275-326
Assaf Naor and Gideon Schechtman, Planar earthmover is not in $L_1$, SIAM J. Comput. 37 (2007), no. 3, 804–826. MR 2341917, DOI 10.1137/05064206X
Aleš Nekvinda and Luděk Zajíček, A simple proof of the Rademacher theorem, Časopis Pěst. Mat. 113 (1988), no. 4, 337–341 (English, with Russian and Czech summaries). MR 981874
I. Protasov, Combinatorics of numbers, Mathematical Studies Monograph Series, vol. 2, VNTL Publishers, L′viv, 1997. Translated from the Russian manuscript by Michael Zarichnyi and Taras Banakh. MR 1723431
Haskell P. Rosenthal, The Banach spaces $C(K)$, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1547–1602. MR 1999603, DOI 10.1016/S1874-5849(03)80043-8
Walter Rudin, Principles of mathematical analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. MR 0385023
Nik Weaver, Lipschitz algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1832645, DOI 10.1142/4100
P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR 1144277, DOI 10.1017/CBO9780511608735