Reflexivity and the fixed-point property for nonexpansive maps

Connections between reflexivity and the fixed-point property for nonexpansive self-mappings of nonempty, closed, bounded, convex subsets of a Banach space are investigated. In particular, it is shown that l(1)(Gamma) for uncountable sets Gamma and l(infinity) cannot even be renormed to have the fixed-point property. As a consequence, if an Orlicz space on a finite measure space that is not purely atomic is endowed with the Orlicz norm, the Orlicz space has the fixed-point property exactly when it is reflexive. (C) 1996 Academic Press, Inc.