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ISSN 1088-6826(e) ISSN 0002-9939(p)

Every nonreflexive subspace of $L_{1}[0,1]$ fails the fixed point property

Author(s): P. N. Dowling; C. J. Lennard
Journal: Proc. Amer. Math. Soc. 125 (1997), 443-446.
MSC (1991): Primary 47H10
MathSciNet review: 1350940
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Abstract: The main result of this paper is that every nonreflexive subspace $Y$ of $L_{1}[0,1]$ fails the fixed point property for closed, bounded, convex subsets $C$ of $Y$ and nonexpansive (or contractive) mappings on $C$ . Combined with a theorem of Maurey we get that for subspaces $Y$ of $L_{1}[0,1]$ , $Y$ is reflexive if and only if $Y$ has the fixed point property. For general Banach spaces the question as to whether reflexivity implies the fixed point property and the converse question are both still open.

References:

[A]: D.E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), 423-424. MR 82j:47070
[CDL]: N.L. Carothers, S.J. Dilworth and C.J. Lennard, On a localization of the UKK property and the fixed point property in $L_{w,1}$ , Interaction Between Functional Analysis, Harmonic Analysis, and Probability (Columbia, MO, 1994), Lecture Notes in Pure and Appl. Math., vol. 175, Dekker, New York, 1966, pp. 111-124.
[D]: J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York Inc., 1984. MR 85i:46020
[DDDL]: P.G. Dodds, T.K. Dodds, P.N. Dowling and C.J. Lennard, Asymptotically isometric copies of $\ell _{1}$ in nonreflexive subspaces of symmetric operator spaces, in preparation.
[DJLT]: P.N. Dowling, W.B. Johnson, C.J. Lennard and B. Turett, The optimality of James's distortion theorems, Proc. Amer. Math. Soc. (to appear).
[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), 653-652.
[ELOS]: J. Elton, P. Lin, E. Odell and S. Szarek, Remarks on the fixed point problem for nonexpansive maps, Contemporary Math., vol. 18, A.M.S., Providence, RI, 1983, pp. 87-119. MR 85d:47059
[KP]: M.I. Kadec and A. Pe{\l}czynski, Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$ , Studia Math. 21 (1962), 161-176. MR 27:2851
[Lim]: T.C. Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math. 90 (1) (1980), 135-143. MR 82h:47052
[Lin]: Pei-Kee Lin, Unconditional bases and fixed points of nonexpansive mappings, Pacific J. Math. 116 (1) (1985), 69-76. MR 86c:46075
[M]: B. Maurey, Points fixes des contractions de certains faiblement compacts de $L^{1}$ , Seminaire d'Analyse Fonctionelle, Exposé no. VIII, École Polytechnique, Centre de Mathé-
matiques (1980-1981). MR 83h:47041
[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.

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Additional Information:

P. N. Dowling
Affiliation: Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056

C. J. Lennard
Affiliation: Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

DOI: 10.1090/S0002-9939-97-03577-6
PII: S 0002-9939(97)03577-6
Keywords: Nonexpansive mapping, contractive mapping, asymptotically isometric copy of $\ell_{1}$, closed, bounded, convex set, fixed point property, nonreflexive subspaces of $L_{1}[0, 1]$
Received by editor(s): March 25, 1995
Received by editor(s) in revised form: August 4, 1995
Communicated by: Dale Alspach
Copyright of article: Copyright 1997, American Mathematical Society

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