A renorming of some nonseparable Banach spaces with the Fixed Point Property

Abstract

We prove that for every Banach space which can be embedded in c 0 ( Γ ) (for instance, reflexive spaces or more generally spaces with M-basis) there exists an equivalent renorming which enjoys the (weak) Fixed Point Property for non-expansive mappings. As a consequence, we solve a longtime open question in Metric Fixed Point Theory: Every reflexive Banach can be renormed to satisfy the Fixed Point Property. Furthermore, this norm can be chosen arbitrarily closed to the original norm.