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Extremal Structure and Duality of Lipschitz Free Spaces

  • Luis García-Lirola
  • Colin Petitjean
  • Antonín Procházka
  • Abraham Rueda Zoca
Article
  • 45 Downloads

Abstract

We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, say x and y, which define it satisfy that the metric segment [xy] only contains x and y. The most notable among them is the case when the free space admits an isometric predual with some additional properties. As an application, we get some new consequences about norm attainment in spaces of vector-valued Lipschitz functions.

Keywords

Extreme point dentability Lipschitz free duality uniformly discrete 

Mathematics Subject Classification

Primary 46B20 Secondary 54E50 

Notes

Acknowledgements

The authors are grateful to Ramón Aliaga and Antonio Guirao for sending them their preprint. The first and the last authors are grateful to the Laboratoire de Mathématiques de Besançon for the excellent working conditions during their visit in June 2017. The third author is grateful to Departamento de Análisis Matemático of Universidad de Granada for hospitality and excellent working conditions during his visit in June 2017. The authors would like to thank Ginés López-Pérez and Matías Raja for useful conversations. Finally, the authors are very grateful to the anonymous referee for their suggestions and their careful reading of the paper.

References

  1. 1.
    Albiac, F., Kalton, N.J.: Lipschitz structure of quasi-Banach spaces. Isr. J. Math. 170, 317–335 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aliaga, R., Guirao, A.J.: On the preserved extremal structure of Lipschitz-free spaces (2017). arXiv:1705.09579
  3. 3.
    Borel-Mathurin, L.: Isomorphismes non linéaires entre espaces de Banach, Ph.D. thesis. Université Paris 6 (2010)Google Scholar
  4. 4.
    Bourgain, J.: On dentability and the Bishop–Phelps property. Isr. J. Math. 28, 265–271 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bourgin, R.D.: Geometric Aspects of Convex Sets with the Radon–Nikodým Property, Lecture Notes in Mathematics, vol. 993. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dalet, A.: Free spaces over countable compact metric spaces. Proc. Am. Math. Soc. 143, 3537–3546 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dalet, A.: Free spaces over some proper metric spaces. Mediterr. J. Math. 12, 973–986 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fabian, M., Habala, P., Hájek, P., Montesinos, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  9. 9.
    Fonf, V.P.: Massiveness of the set of extremal points of the unit sphere of some conjugate Banach spaces. Ukrain. Mat. Zh 30, 846–849 (1978). 863MathSciNetGoogle Scholar
  10. 10.
    García-Lirola, L.: Convexity, optimization and geometry of the ball in Banach spaces, Ph.D. thesis. Universidad de Murcia (2017)Google Scholar
  11. 11.
    García-Lirola, L., Petitjean, C., Rueda Zoca, A.: On the structure of spaces of vector-valued Lipschitz functions. Studia Math. 239, 249–271 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    García-Lirola, L., Procházka, A., Rueda Zoca, A.: A characterisation of the Daugavet property in spaces of Lipschitz functions (2017). arXiv:1705.05145
  13. 13.
    García-Lirola, L., Rueda Zoca, A.: Unconditional almost squareness and applications to spaces of Lipschitz functions. J. Math. Anal. Appl. 451, 117–131 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Godefroy, G.: Boundaries of a convex set and interpolation sets. Math. Ann. 277, 173–184 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Godefroy, G.: A survey on Lipschitz-free Banach spaces. Comment. Math. 55, 89–118 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Guirao, A.J., Montesinos, V., Zizler, V.: On preserved and unpreserved extreme points. In: Descriptive Topology and Functional Analysis, Springer Proc. Math. Stat., vol. 80. Springer, Cham, pp. 163–193 (2014)Google Scholar
  17. 17.
    Ivakhno, Y., Kadets, V., Werner, D.: Corrigendum to: The Daugavet property for spaces of Lipschitz functions. Math. Scand. 104, 319 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kadets, V., Martín, M., Soloviova, M.: Norm-attaining Lipschitz functionals. Banach J. Math. Anal. 10, 621–637 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kalton, N.J.: Spaces of Lipschitz and Hölder functions and their applications. Collect. Math. 55, 171–217 (2004)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Matoušková, E.: Extensions of continuous and Lipschitz functions. Can. Math. Bull. 43, 208–217 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Petitjean, C.: Lipschitz-free spaces and Schur properties. J. Math. Anal. Appl. 453, 894–907 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Petunīn, J.I., Plīčko, A.N.: Some properties of the set of functionals that attain a supremum on the unit sphere. Ukrain. Mat. Ž. 26, 102–106, 143 (1974)Google Scholar
  23. 23.
    Ryan, R.A.: Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics. Springer, London (2002)CrossRefGoogle Scholar
  24. 24.
    Weaver, N.: Lipschitz Algebras. World Scientific, River Edge (1999)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Facultad de MatemáticasUniversidad de MurciaEspinardoSpain
  2. 2.Laboratoire de Mathématiques UMR 6623Université Bourgogne Franche-ComtéBesançon CedexFrance
  3. 3.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain

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