Spaces not containing ${\ell }_1$ have weak aproximate fixed point property

Abstract

A nonempty closed convex bounded subset $C$ of a Banach space is said to have the weak approximate fixed point property if for every continuous map $f:C\to C$ there is a sequence $\{x_n\}$ in $C$ such that $x_n-f(x_n)$ converge weakly to 0. We prove in particular that $C$ has this property whenever it contains no sequence equivalent to the standard basis of ${\ell }_1$. As a byproduct we obtain a characterization of Banach spaces not containing ${\ell }_1$ in terms of the weak topology.

6 pages; the paper was reorganized a bit