Embedding $\ell _1$ as Lipschitz functions
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Abstract:
Let $K$ be a compact Hausdorff space and let $d$ be a lower semicontinuous metric on it. We prove that $K$ is fragmented by $d$ if, and only if, $C(K)$ contains no copy of $\ell _1$ made up of Lipschitz functions with respect to $d$. As applications we obtain a characterization of Asplund Banach spaces and Radon-Nikodým compacta.References
- Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
- B. Cascales, I. Namioka, and J. Orihuela, The Lindelöf property in Banach spaces, Studia Math. 154 (2003), no. 2, 165–192. MR 1949928, DOI 10.4064/sm154-2-4
- Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
- Gustave Choquet, Topology, Pure and Applied Mathematics, Vol. XIX, Academic Press, New York-London, 1966. Translated from the French by Amiel Feinstein. MR 0193605
- Robert C. James, A separable somewhat reflexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80 (1974), 738–743. MR 417763, DOI 10.1090/S0002-9904-1974-13580-9
- J. E. Jayne, I. Namioka, and C. A. Rogers, Norm fragmented weak${}^\ast$ compact sets, Collect. Math. 41 (1990), no. 2, 133–163 (1991). MR 1149650
- J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain $\ell _{1}$ and whose duals are non-separable, Studia Math. 54 (1975), no. 1, 81–105. MR 390720, DOI 10.4064/sm-54-1-81-105
- Eva Matoušková, Extensions of continuous and Lipschitz functions, Canad. Math. Bull. 43 (2000), no. 2, 208–217. MR 1754025, DOI 10.4153/CMB-2000-028-0
- Paul R. Meyer, The Baire order problem for compact spaces, Duke Math. J. 33 (1966), 33–39. MR 190897
- I. Namioka, Radon-Nikodým compact spaces and fragmentability, Mathematika 34 (1987), no. 2, 258–281. MR 933504, DOI 10.1112/S0025579300013504
- Haskell P. Rosenthal, A characterization of Banach spaces containing $l^{1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. MR 358307, DOI 10.1073/pnas.71.6.2411
- Stevo Todorcevic, Topics in topology, Lecture Notes in Mathematics, vol. 1652, Springer-Verlag, Berlin, 1997. MR 1442262, DOI 10.1007/BFb0096295
Additional Information
- M. Raja
- Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel
- Address at time of publication: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain
- Email: matias@um.es
- Received by editor(s): March 23, 2004
- Published electronically: March 15, 2005
- Additional Notes: This research was supported by a grant of Professor J. Lindenstrauss from the Israel Science Foundation, and by research grant BFM2002-01719, MCyT (Spain).
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 2395-2400
- MSC (2000): Primary 46B20, 46B22, 54E99
- DOI: https://doi.org/10.1090/S0002-9939-05-07943-8
- MathSciNet review: 2138882