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

Authors: P. N. Dowling and 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.

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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

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
Article copyright: © Copyright 1997 American Mathematical Society