Lipschitz-quotients and the Kunen-Martin Theorem

Yves Dutrieux

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 4, page 641-648
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Dutrieux, Yves. "Lipschitz-quotients and the Kunen-Martin Theorem." Commentationes Mathematicae Universitatis Carolinae 42.4 (2001): 641-648. <http://eudml.org/doc/248784>.

@article{Dutrieux2001,
abstract = {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.},
author = {Dutrieux, Yves},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lipschitz equivalences; Szenk index; Lipschitz equivalences; Szlenk index; descriptive set theory; Banach space},
language = {eng},
number = {4},
pages = {641-648},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Lipschitz-quotients and the Kunen-Martin Theorem},
url = {http://eudml.org/doc/248784},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Dutrieux, Yves
TI - Lipschitz-quotients and the Kunen-Martin Theorem
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 4
SP - 641
EP - 648
AB - 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.
LA - eng
KW - Lipschitz equivalences; Szenk index; Lipschitz equivalences; Szlenk index; descriptive set theory; Banach space
UR - http://eudml.org/doc/248784
ER -

References

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  2. 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) Zbl0954.46014MR1736929
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  9. 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) MR1222962
  10. Benyamini J., Lindenstrauss Y., Geometric nonlinear functional analysis vol. I, AMS Colloquium Publications (2000), 48. (2000) 
  11. Godefroy N., Kalton G., Lancien G., Subspaces of c 0 ( ℕ ) and Lipschitz isomorphisms, Geom. Funct. Anal. (2000), to appear. 
  12. Ribe M, Existence of separable uniformly homeomorphic non isomorphic Banach spaces, Israel J. Math. (1984), 48 139-147. (1984) MR0770696
  13. Bessaga A., Pełczyński C., Spaces of continuous functions (IV), Studia Math. (1960), 19 53-62. (1960) Zbl0094.30303MR0113132

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