Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on asymptotically isometric copies of $l^1$ and $c_0$

Author: Hermann Pfitzner
Journal: Proc. Amer. Math. Soc. 129 (2001), 1367-1373
MSC (2000): Primary 46B03, 46B04, 46B20, 47H10
Published electronically: October 20, 2000
MathSciNet review: 1814162
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Nonreflexive Banach spaces that are complemented in their bidual by an L-projection--like preduals of von Neumann algebras or the Hardy space $H^1$--contain, roughly speaking, many copies of $l^1$ which are very close to isometric copies. Such $l^1$-copies are known to fail the fixed point property. Similar dual results hold for $c_0$.

References [Enhancements On Off] (What's this?)

  • 1. D. E. Alspach.
    A fixed point free nonexpansive map.
    Proc. Amer. Math. Soc., 82:423-424, 1981. MR 82j:47070
  • 2. A. V. Buhvalov and G. J. Lozanovskii.
    On sets closed in measure in spaces of measurable functions.
    Trans. Moscow Math. Soc., 2:127-148, 1978.
  • 3. J. Diestel.
    Sequences and Series in Banach Spaces.
    Springer, Berlin-Heidelberg-New York, 1984. MR 85i:46020
  • 4. P. N. Dowling, W. B. Johnson, C. J. Lennard, and B. Turett.
    The optimality of James's distortion theorems.
    Proc. Amer. Math. Soc., 125:167-174, 1997. MR 97d:46010
  • 5. P. N. Dowling and C. J. Lennard.
    Every nonreflexive subspace of $L_1[0,1]$ fails the fixed point property.
    Proc. Amer. Math. Soc., 125:443-446, 1997. MR 97d:46034
  • 6. P. N. Dowling, C. J. Lennard, and B. Turett.
    Reflexivity and the Fixed-Point-Property for Nonexpansive Maps.
    J. Math. Anal. Appl., 200:653-662, 1996. MR 97c:47062
  • 7. G. Godefroy.
    Sous-espaces bien disposés de $L^{1}$ - Applications.
    Trans. Amer. Math. Soc., 286:227-249, 1984. MR 86h:46033
  • 8. P. Harmand, D. Werner, and W. Werner.
    $M$-ideals in Banach Spaces and Banach Algebras.
    Lecture Notes in Mathematics 1547. Springer, 1993. MR 94k:46022
  • 9. R. C. James.
    Uniformly non-square Banach spaces.
    Ann. of Math., 80:542-550, 1964. MR 30:4139
  • 10. I. Kadec and A. Pe\lczynski.
    Bases, lacunary sequences and complemented subspaces in the space ${\mbox{L}}_{p}$.
    Studia Math., 21:161-176, 1962.
  • 11. D. Li.
    Espaces $L$-facteurs de leurs biduaux: bonne disposition, meilleure approximation et propriété de Radon-Nikodym.
    Quart. J. Math. Oxford (2), 38:229-243, 1987. MR 88h:46024
  • 12. J. Lindenstrauss and L. Tzafriri.
    Classical Banach Spaces I.
    Springer, Berlin-Heidelberg-New York, 1977. MR 58:17766
  • 13. J. Lindenstrauss and L. Tzafriri.
    Classical Banach Spaces II.
    Springer, Berlin-Heidelberg-New York, 1979. MR 81c:46001
  • 14. H. Pfitzner.
    L-summands in their biduals have Penski's property (V$^*$).
    Studia Math., 104:91-98, 1993. MR 94f:46021

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B03, 46B04, 46B20, 47H10

Retrieve articles in all journals with MSC (2000): 46B03, 46B04, 46B20, 47H10

Additional Information

Hermann Pfitzner
Affiliation: Université d’Orléans, BP 6759, F-45067 Orléans Cedex 2, France

Keywords: Asymptotically isometric copies of $l_1$, James' distortion, L-summands, L-embedded, M-ideals, fixed point property
Received by editor(s): July 20, 1999
Published electronically: October 20, 2000
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society