THE FIXED POINT PROPERTY VIA DUAL SPACE PROPERTIES

Abstract. A Banach space has the weak fixed point property if its dual space has a weak ∗ sequentially compact unit ball and the dual space satisfies the weak ∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε> 0 such that, for every infinite subset A of the unit sphere of the dual space, A ∪ (−A) fails to be (2 − ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property. Determining conditions on a Banach space X so that every nonexpansive mapping from a nonempty, closed, bounded, convex subset of X into itself has a fixed point has been of considerable interest for many years. A Banach space has the fixed point property if, for each nonempty, closed, bounded, convex subset C of X, every nonexpansive mapping of C into itself has a fixed point. A Banach space is said to have the weak fixed point property if the class of sets C above is restricted to the set of weakly compact convex sets; and a Banach space is said to have the weak ∗ fixed point property if X is a dual space and the class of sets C is restricted to the set o