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
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.
B. Brosowski and F. Deutsch, On some geometric properties of suns, J. Approximation Theory 10 (1974), 245–267.
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.
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.
R. Durier and P. L. Papini, Polyhedral norms in an infinite-dimensional space, Rocky Mountain J. Math. 23 (1993), 863–875.
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.
V. P. Fonf and L. Veselý, Infinite-dimensional polyhedrality, Canad. J. Math. 56 (2004), 472–494.
I. Gasparis, On contractively complemented subspaces of separable L1-preduals, Israel J. Math. 128 (2002), 77–92.
A. Gleit and R. McGuigan, A note on polyhedral Banach spaces, Proc. Amer. Math. Soc. 33 (1972), 398–404.
V. Klee, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79–107.
V. Klee, Polyhedral sections of convex bodies, Acta Math. 103 (1960), 243–267.
H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer-Verlag, New York-Heidelberg, 1974, Die Grundlehren der mathematischen Wissenschaften, Band 208.
A. J. Lazar, Polyhedral Banach spaces and extensions of compact operators, Israel J. Math. 7 (1969), 357–364.
J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. No. 48 (1964), 112.
A. J. Lazar and J. Lindenstrauss, On Banach spaces whose duals are L1 spaces, Israel J. Math. 4 (1966), 205–207.
Z. Semadeni, Free compact convex sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 141–146.
L. Veselý, Boundary of polyhedral spaces: an alternative proof, Extracta Math. 15 (2000), 213–217.
M. Zippin, On some subspaces of Banach spaces whose duals are L1 spaces, Proc. Amer. Math. Soc. 23 (1969), 378–385.
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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