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Rethinking polyhedrality for lindenstrauss spaces

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Abstract

We present a Lindenstrauss space with an extreme point that does not contain a subspace linearly isometric to c. This example disproves a result stated by Zippin in a paper published in 1969 and it shows that some classical characterizations of polyhedral Lindenstrauss spaces, based on Zippin’s result, are false, whereas some others remain unproven; then we provide a correct proof for those characterizations. Finally, we also disprove a characterization of polyhedral Lindenstrauss spaces given by Lazar, in terms of the compact norm-preserving extension of compact operators, and we give an equivalent condition for a Banach space X to satisfy this property.

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References

  1. D. E. Alspach, A l1-predual which is not isometric to a quotient of C(a), in Banach spaces (Mérida, 1992), Contemp. Math., Vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 9–14.

    Chapter  Google Scholar 

  2. B. Brosowski and F. Deutsch, On some geometric properties of suns, J. Approximation Theory 10 (1974), 245–267.

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Casini, E. Miglierina and L. Piasecki, Hyperplanes in the space of convergent sequences and preduals of 1, Canad. Math. Bull. 58 (2015), 459–470.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Casini, E. Miglierina and L. Piasecki, Separable Lindenstrauss spaces whose duals lack the weak* fixed point property for nonexpansive mappings, arXiv:1503.08875 [math.FA], March 2015.

    Google Scholar 

  5. R. Durier and P. L. Papini, Polyhedral norms in an infinite-dimensional space, Rocky Mountain J. Math. 23 (1993), 863–875.

    Article  MathSciNet  MATH  Google Scholar 

  6. V. P. Fonf, J. Lindenstrauss and R. R. Phelps, Infinite Dimensional Convexity, in Handbook of the geometry of Banach spaces, Vol. I (W. B. Johnson and J. Lindenstrauss, eds.), North-Holland, Amsterdam, 2001, pp. 599–670.

  7. V. P. Fonf and L. Veselý, Infinite-dimensional polyhedrality, Canad. J. Math. 56 (2004), 472–494.

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Gasparis, On contractively complemented subspaces of separable L1-preduals, Israel J. Math. 128 (2002), 77–92.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Gleit and R. McGuigan, A note on polyhedral Banach spaces, Proc. Amer. Math. Soc. 33 (1972), 398–404.

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Klee, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79–107.

    Article  MathSciNet  MATH  Google Scholar 

  11. V. Klee, Polyhedral sections of convex bodies, Acta Math. 103 (1960), 243–267.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer-Verlag, New York-Heidelberg, 1974, Die Grundlehren der mathematischen Wissenschaften, Band 208.

    Book  MATH  Google Scholar 

  13. A. J. Lazar, Polyhedral Banach spaces and extensions of compact operators, Israel J. Math. 7 (1969), 357–364.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. No. 48 (1964), 112.

    MathSciNet  MATH  Google Scholar 

  15. A. J. Lazar and J. Lindenstrauss, On Banach spaces whose duals are L1 spaces, Israel J. Math. 4 (1966), 205–207.

    Article  MathSciNet  MATH  Google Scholar 

  16. Z. Semadeni, Free compact convex sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 141–146.

    MathSciNet  MATH  Google Scholar 

  17. L. Veselý, Boundary of polyhedral spaces: an alternative proof, Extracta Math. 15 (2000), 213–217.

    MathSciNet  MATH  Google Scholar 

  18. M. Zippin, On some subspaces of Banach spaces whose duals are L1 spaces, Proc. Amer. Math. Soc. 23 (1969), 378–385.

    MathSciNet  MATH  Google Scholar 

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Correspondence to Emanuele Casini.

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Casini, E., Miglierina, E., Piasecki, Ł. et al. Rethinking polyhedrality for lindenstrauss spaces. Isr. J. Math. 216, 355–369 (2016). https://doi.org/10.1007/s11856-016-1412-8

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  • DOI: https://doi.org/10.1007/s11856-016-1412-8

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