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)

 

 

The optimality of James's distortion theorems


Authors: P. N. Dowling, W. B. Johnson, C. J. Lennard and B. Turett
Journal: Proc. Amer. Math. Soc. 125 (1997), 167-174
MSC (1991): Primary 46B03, 46B20
DOI: https://doi.org/10.1090/S0002-9939-97-03537-5
MathSciNet review: 1346969
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A renorming of $\ell _{1}$, explored here in detail, shows that the copies of $\ell _{1}$ produced in the proof of the Kadec-Pelczynski theorem inside nonreflexive subspaces of $L_{1}[0,1]$ cannot be produced inside general nonreflexive spaces that contain copies of $\ell _{1}$. Put differently, James's distortion theorem producing one-plus-epsilon-isomorphic copies of $\ell _{1}$ inside any isomorphic copy of $\ell _{1}$ is, in a certain sense, optimal. A similar renorming of $c_{0}$ shows that James's distortion theorem for $c_{0}$ is likewise optimal.


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

  • [BP] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 0115069
  • [CDL] N. L. Carothers, S. J. Dilworth and C. J. Lennard, On a localization of the UKK property and the fixed point property in $L_{w,1}$, Lecture Notes in Pure and Appl. Math., vol. 175, Dekker, New York, 1996, pp. 111-124. CMP 96:03
  • [D] Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004
  • [DL] P. N. Dowling and C. J. Lennard, Every nonreflexive subspace of $L_{1}[0,1]$ fails the fixed point property, Proc. Amer. Math. Soc. (to appear). CMP 96:01
  • [DLT] P. N. Dowling, C. J. Lennard and B. Turett, Reflexivity and the fixed point property for nonexpansive maps, J. Math. Anal. Appl. 200 (1996), 653-662.
  • [J] Robert C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542–550. MR 0173932, https://doi.org/10.2307/1970663
  • [KP] M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces 𝐿_{𝑝}, Studia Math. 21 (1961/1962), 161–176. MR 0152879
  • [LT] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056
  • [M] B. Maurey, Points fixes des contractions de certains faiblement compacts de 𝐿¹, Seminar on Functional Analysis, 1980–1981, École Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19 (French). MR 659309
  • [S] M. Smyth, Remarks on the weak star fixed point property in the dual of $C(\Omega )$, J. Math. Anal. Appl. 195 (1995), 294-306. CMP 96:01

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46B03, 46B20

Retrieve articles in all journals with MSC (1991): 46B03, 46B20


Additional Information

P. N. Dowling
Affiliation: Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056
Email: pndowling@miavx1.acs.muohio.edu

W. B. Johnson
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: johnson@math.tamu.edu

C. J. Lennard
Affiliation: Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: chris@lennext.math.pitt.edu

B. Turett
Affiliation: Department of Mathematical Sciences, Oakland University, Rochester, Michigan 48309
Email: turett@vela.acs.oakland.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03537-5
Keywords: $\ell _{1}$, $c_{0}$, renorming, James's distortion theorem, asymptotically isometric copies of $\ell _{1}$, fixed point property
Received by editor(s): May 8, 1995
Received by editor(s) in revised form: July 7, 1995
Additional Notes: The second author was supported by NSF 93-06376.
The third author was partially supported by a University of Pittsburgh FAS grant.
Communicated by: Dale Alspach
Article copyright: © Copyright 1997 American Mathematical Society