Every nonreflexive subspace of

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

DOI:
https://doi.org/10.1090/S0002-9939-97-03577-6

MathSciNet review:
1350940

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Abstract | References | Similar Articles | Additional Information

Abstract: The main result of this paper is that every nonreflexive subspace of fails the fixed point property for closed, bounded, convex subsets of and nonexpansive (or contractive) mappings on . Combined with a theorem of Maurey we get that for subspaces of , is reflexive if and only if 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

DOI:
https://doi.org/10.1090/S0002-9939-97-03577-6

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

Article copyright:
© Copyright 1997
American Mathematical Society