Szlenk indices and uniform homeomorphisms

Godefroy, G.; Kalton, N.; Lancien, G.

doi:10.1090/S0002-9947-01-02825-2

Szlenk indices and uniform homeomorphisms
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by G. Godefroy, N. J. Kalton and G. Lancien

Trans. Amer. Math. Soc. 353 (2001), 3895-3918

DOI: https://doi.org/10.1090/S0002-9947-01-02825-2

Published electronically: May 17, 2001

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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.

References

Dale Alspach, The dual of the Bourgain-Delbaen space, Israel J. Math. 117 (2000), 239–259. MR 1760594, DOI 10.1007/BF02773572
D. Alspach, R. Judd and E. Odell, The Szlenk index and local $l_1$ indices of a Banach space, to appear.
Texas functional analysis seminar 1984–1985, Longhorn Notes, University of Texas Press, Austin, TX, 1985. MR 832245
S. J. Dilworth, Maria Girardi, and Denka Kutzarova, Banach spaces which admit a norm with the uniform Kadec-Klee property, Studia Math. 112 (1995), no. 3, 267–277. MR 1311702
Y. Dutrieux, Quotients of $c_0$ and Lipschitz-homeomorphisms, Houston J. Math., to appear.
Per Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972), 281–288 (1973). MR 336297, DOI 10.1007/BF02762802
G. Godefroy, N.J. Kalton and G. Lancien, Subspaces of $c_0(\mathbb N)$ and Lipschitz isomorphisms, Geom. Funct. Anal. 10 (2000) 798-820.
Gilles Godefroy, Nigel J. Kalton, and Gilles Lancien, L’espace de Banach $c_0$ est déterminé par sa métrique, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 9, 817–822 (French, with English and French summaries). MR 1663734, DOI 10.1016/S0764-4442(99)80111-6
E. Gorelik, The uniform nonequivalence of $L_p$ and $l_p$, Israel J. Math. 87 (1994), no. 1-3, 1–8. MR 1286810, DOI 10.1007/BF02772978
R. Haydon, Subspaces of the Bourgain-Delbaen space, Studia Math. 139 (2000) 275-293.
S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), no. 3, 225–251. MR 675426, DOI 10.4064/sm-73-3-225-251
W. B. Johnson, On quotients of $L_{p}$ which are quotients of $l_{p}$, Compositio Math. 34 (1977), no. 1, 69–89. MR 454595
W.B. Johnson, J. Lindenstrauss, D. Preiss and G. Schechtman, Almost Fréchet differentiability of Lipschitz mappings between infinite dimensional Banach spaces, Preprint (2000).
W. B. Johnson, J. Lindenstrauss, and G. Schechtman, Banach spaces determined by their uniform structures, Geom. Funct. Anal. 6 (1996), no. 3, 430–470. MR 1392325, DOI 10.1007/BF02249259
W. B. Johnson and E. Odell, Subspaces of $L_{p}$ which embed into $l_{p}$, Compositio Math. 28 (1974), 37–49. MR 352938
Nigel J. Kalton and Mikhail I. Ostrovskii, Distances between Banach spaces, Forum Math. 11 (1999), no. 1, 17–48. MR 1673915, DOI 10.1515/form.11.1.17
Nigel J. Kalton and Dirk Werner, Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137–178. MR 1324212, DOI 10.1515/crll.1995.461.137
H. Knaust, E. Odell and T. Schlumprecht, On asymptotic structure, the Szlenk index and UKK properties in Banach spaces, Positivity 3 (1999) 173-199.
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.
Gilles Lancien, On uniformly convex and uniformly Kadec-Klee renormings, Serdica Math. J. 21 (1995), no. 1, 1–18. MR 1333523
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.
D. R. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to $l_{1}(\Gamma )$, J. Functional Analysis 12 (1973), 177–187. MR 0342987, DOI 10.1016/0022-1236(73)90022-0
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
V. D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspehi Mat. Nauk 26 (1971), no. 6(162), 73–149 (Russian). MR 0420226
E. Odell and H. P. Rosenthal, A double-dual characterization of separable Banach spaces containing $l^{1}$, Israel J. Math. 20 (1975), no. 3-4, 375–384. MR 377482, DOI 10.1007/BF02760341
Gilles Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), no. 3-4, 326–350. MR 394135, DOI 10.1007/BF02760337
M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces, Israel J. Math. 48 (1984), no. 2-3, 139–147. MR 770696, DOI 10.1007/BF02761159
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 227743, DOI 10.4064/sm-30-1-53-61