Every nonreflexive subspace of ๐ฟโ[0,1] fails the fixed point property
Patrick N. Dowling, Chris Lennard ยท Proceedings of the American Mathematical Society ยท 1997
The main result of this paper is that every nonreflexive subspace Y Y of L 1 [ 0 , 1 ] L_{1}[0,1] fails the fixed point property for closed, bounded, convex subsets C C of Y Y and nonexpansive (or contractive) mappings on C C . Combined with a theorem of Maurey we get that for subspaces Y Y of L 1 [ 0 , 1 ] L_{1}[0,1] , Y Y is reflexive if and only if Y 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.