Proceedings of the American Mathematical Society

Author(s): P. N. Dowling; C. J. Lennard
Journal: Proc. Amer. Math. Soc. 125 (1997), 443-446.
MSC (1991): Primary 47H10
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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 $C$

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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Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47H10

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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