Lipschitz-quotients and the Kunen-Martin Theorem
Dutrieux, Yves
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001), p. 641-648 / Harvested from Czech Digital Mathematics Library
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We show that there is a universal control on the Szlenk index of a Lipschitz-quotient of a Banach space with countable Szlenk index. It is in particular the case when two Banach spaces are Lipschitz-homeomorphic. This provides information on the Cantor index of scattered compact sets $K$ and $L$ such that $C(L)$ is a Lipschitz-quotient of $C(K)$ (that is the case in particular when these two spaces are Lipschitz-homeomorphic). The proof requires tools of descriptive set theory.

Publié le : 2001-01-01
Classification:  03E15,  46B20 
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     author = {Yves Dutrieux},
     title = {Lipschitz-quotients and the Kunen-Martin Theorem},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {42},
     year = {2001},
     pages = {641-648},
     zbl = {1069.03035},
     mrnumber = {1883373},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119280}
}

Dutrieux, Yves. Lipschitz-quotients and the Kunen-Martin Theorem. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 641-648. http://gdmltest.u-ga.fr/item/119280/

Godefroy N.J.; Kalton G.; Lancien G. Szlenk indices and uniform homeomorphisms, Trans. Amer. Math. Soc. (2000), to appear. | MR 1837213 | Zbl 0981.46007

Bates W.; Johnson J.; Lindenstrauss D.; Preiss G.; Schechtman S. Affine approximation of Lipschitz functions and nonlinear quotients, Geom. Funct. Anal. (1999), 6 1092-1127. (1999) | MR 1736929 | Zbl 0954.46014

Bossard B. Théorie descriptive des ensembles en géométrie des espaces de Banach, Thèse de doctorat de l'Université Paris VI, (in French), 1994.

Lancien G. On the Szlenk index and the weak*-dentability index, Quart. J. Math. Oxford (2) (1996), 47 59-71. (1996) | MR 1380950 | Zbl 0973.46014

Kechris A.; Louveau A. Descriptive set theory and the structure of sets of uniqueness, London Math. Soc. Lecture Note Series (1987), 128. (1987) | Zbl 0642.42014

Heinrich P.; Mankiewicz S. Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. (1982), 73 225-251. (1982) | MR 0675426 | Zbl 0506.46008

Bourgain J. New classes of $\Cal L^p$-spaces, Lecture Notes in Mathematics 889, Springer Verlag, 1981. | MR 0639014

Christensen J. Topology and Borel structure; descriptive topology and set theory with applications to functional analysis and measure theory, North-Holland Math. Stud. (1974), 10. (1974) | MR 0348724 | Zbl 0273.28001

Bossard B. Codages des espaces de Banach séparables; familles analytiques et coanalytiques d'espaces de Banach, C.R. Acad. Sci. Paris, Série I, 316 (1993), 1005-1010. (1993) | MR 1222962

Benyamini J.; Lindenstrauss Y. Geometric nonlinear functional analysis vol. I, AMS Colloquium Publications (2000), 48. (2000)

Godefroy N.; Kalton G.; Lancien G. Subspaces of $c_0(\Bbb N)$ and Lipschitz isomorphisms, Geom. Funct. Anal. (2000), to appear.

Ribe M Existence of separable uniformly homeomorphic non isomorphic Banach spaces, Israel J. Math. (1984), 48 139-147. (1984) | MR 0770696

Bessaga A.; Pełczyński C. Spaces of continuous functions (IV), Studia Math. (1960), 19 53-62. (1960) | MR 0113132 | Zbl 0094.30303