Spaces not containing ℓ1 have weak approximate fixed point property

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→C there is a sequence {xn} in C such that xn−f(xn) converge weakly to 0. We prove in particular that C has this property whenever it contains no sequence equivalent to the standard basis of ℓ1. As a byproduct we obtain a characterization of Banach spaces not containing ℓ1 in terms of the weak topology.