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Szlenk indices and uniform homeomorphisms

Authors: G. Godefroy, N. J. Kalton and G. Lancien
Journal: Trans. Amer. Math. Soc. 353 (2001), 3895-3918
MSC (2000): Primary 46B03, 46B20
Published electronically: May 17, 2001
MathSciNet review: 1837213
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Abstract: We prove some rather precise renorming theorems for Banach spaces with Szlenk index $\omega_0.$ We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.

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  • 1. D. Alspach, The dual of the Bourgain-Delbaen space, Israel J. Math. 117 (2000) 239-259. MR 2001d:46022
  • 2. D. Alspach, R. Judd and E. Odell, The Szlenk index and local $l_1$ indices of a Banach space, to appear.
  • 3. Y. Benyamini, The uniform classification of Banach spaces, Longhorn Notes, University of Texas Austin, 1984-5, pp. 15-38. MR 87a:46007
  • 4. S.J. Dilworth, M. Girardi and D. Kutzarova, Banach spaces which admit a norm with the uniform Kadec-Klee property, Studia Math. 112 (1995) 267-277. MR 96a:46023
  • 5. Y. Dutrieux, Quotients of $c_0$ and Lipschitz-homeomorphisms, Houston J. Math., to appear.
  • 6. P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972) 281-288. MR 49:1073
  • 7. G. Godefroy, N.J. Kalton and G. Lancien, Subspaces of $c_0(\mathbb N)$ and Lipschitz isomorphisms, Geom. Funct. Anal. 10 (2000) 798-820. CMP 2001:03
  • 8. G. Godefroy, N.J. Kalton and G. Lancien, L'espace de Banach $c_0$ est déterminé par sa métrique, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 817-822. MR 99h:46007
  • 9. E. Gorelik, The uniform non-equivalence of $L_p$ and $\ell_p$, Israel J. Math. 87 (1994) 1-8. MR 95f:46028
  • 10. R. Haydon, Subspaces of the Bourgain-Delbaen space, Studia Math. 139 (2000) 275-293. CMP 2000:13
  • 11. S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982) 225-251. MR 84h:46026
  • 12. W.B. Johnson, On quotients of $L^p$ which are quotients of $l^p$, Compositio Mathematica, 34 (1977) 69-89. MR 56:12844
  • 13. W.B. Johnson, J. Lindenstrauss, D. Preiss and G. Schechtman, Almost Fréchet differentiability of Lipschitz mappings between infinite dimensional Banach spaces, Preprint (2000).
  • 14. W.B. Johnson, J. Lindenstrauss and G. Schechtman, Banach spaces determined by their uniform structure, Geom. Funct. Anal. 3 (1996) 430-470. MR 97b:46016
  • 15. W.B. Johnson and E. Odell, Subspaces of $L_p$ which embed into $\ell_p$, Compositio Math. 28 (1974) 37-49. MR 50:5424
  • 16. N.J. Kalton and M.M. Ostrovskii, Distances between Banach spaces, Forum Math. 11(1999), 17-48. MR 2000c:46024
  • 17. N.J. Kalton and D. Werner, Property (M), M-ideals and almost isometric structure, J. Reine Angew. Math. 461 (1995) 137-178. MR 96m:46022
  • 18. H. Knaust, E. Odell and T. Schlumprecht, On asymptotic structure, the Szlenk index and UKK properties in Banach spaces, Positivity 3 (1999) 173-199. CMP 99:16
  • 19. G. Lancien, Théorie de l'indice et problèmes de renormage en géometrie des espaces de Banach, Thèse de doctorat de l'Université Paris VI, January 1992.
  • 20. G. Lancien, On uniformly convex and uniformly Kadec-Klee renormings, Serdica Math. J. 21 (1995) 1-18. MR 96e:46009
  • 21. G. Lancien, Réflexivité et normes duales possédant la propriété uniforme de Kadec-Klee, Publ. Math. de la Fac. des sciences de Besançon, Fasicule 14 (1993-94) 69-74.
  • 22. D.R. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to $\ell_1(\Gamma),$ J. Functional Analysis 12 (1973) 177-187. MR 49:7731
  • 23. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Vol. 1, Sequence spaces, Springer-Verlag, 1977. MR 58:17766
  • 24. V. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball (Russian) Uspehi Mat. Nauk 26 (1971), 6 (162), 73-149. English translation: Russian Math. Surveys 26 (1971), 6, 79-163. MR 54:8240
  • 25. E. Odell and H.P. Rosenthal, A double dual characterization of separable Banach spaces containing $\ell1,$ Israel J. Math. 20 (1975) 375-384. MR 51:13654
  • 26. G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326-350. MR 52:14940
  • 27. M. Ribe, Existence of separable uniformly homeomorphic non-isomorphic Banach spaces, Israel J. Math. 48 (1984), 139-147. MR 86e:46015
  • 28. W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia Math. 30 (1968) 53-61. MR 37:3327

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Additional Information

G. Godefroy
Affiliation: Equipe d’Analyse, Université Paris VI, Boite 186, 4, Place Jussieu, 75252 Paris Cedex 05, France

N. J. Kalton
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211

G. Lancien
Affiliation: Equipe de Mathématiques - UMR 6623, Université de Franche-Comté, F-25030 Besançon cedex

Received by editor(s): June 15, 1999
Received by editor(s) in revised form: July 3, 2000
Published electronically: May 17, 2001
Additional Notes: The second author was supported by NSF grant DMS-9870027.
Article copyright: © Copyright 2001 American Mathematical Society

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