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

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Abstract | References | Similar Articles | Additional Information

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

**[BP]**C. Bessaga and A. Pelczynski,*On bases and unconditional convergence of series in Banach spaces*, Studia Math.**17**(1958), 151-164. MR**22:5872****[CDL]**N. L. Carothers, S. J. Dilworth and C. J. Lennard,*On a localization of the UKK property and the fixed point property in*, Lecture Notes in Pure and Appl. Math., vol. 175, Dekker, New York, 1996, pp. 111-124. CMP**96:03****[D]**J. Diestel,*Sequences and Series in Banach Spaces*, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984. MR**85i:46020****[DL]**P. N. Dowling and C. J. Lennard,*Every nonreflexive subspace of 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]**R. C. James,*Uniformly non-square Banach spaces*, Ann. of Math.**80**(1964), 542-550. MR**30:4139****[KP]**M. I. Kadec and A. Pelczynski,*Bases, lacunary sequences and complemented subspaces in*, Studia Math.**21**(1962), 161-176. MR**27:2851****[LT]**J. Lindenstrauss and L. Tzafriri,*Classical Banach Spaces I : Sequence Spaces*, Springer-Verlag, Berlin, Heidelberg, New York, 1977. MR**58:17766****[M]**B. Maurey,*Points fixes des contractions de certains faiblement compacts de*, Seminaire d' Analyse Fonctionelle, Exposé no. VIII, École Polytechnique, Centre de Mathématiques (1980-1981). MR**83h:47041****[S]**M. Smyth,*Remarks on the weak star fixed point property in the dual of*, J. Math. Anal. Appl.**195**(1995), 294-306. CMP**96:01**

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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