Dual fixed point property on spaces of continuous functions under equivalent norms

Abstract

A Banach space X is said to have the fixed point property if for every closed convex bounded subset C⊂X and for every T:C→C nonexpansive, there is a fixed point. A Banach space is said to have the dual-fixed point property if its dual space X⁎ verifies the fixed point property. It is a fact that (C0(K),‖⋅‖∞) fails the dual-fixed point property for every locally compact Hausdorff set K. In this article we study the fulfilment of the dual-fixed point property under equivalent norms. We prove that C0(K) can be renormed to have the dual-fixed point property if and only if K is a countable set. In this case we prove that this renorming can be achieved as close to the standard norm on C0(K) as we like for the Banach-Mazur distance. We further obtain the corresponding weak⁎-fixed-point-property stability coefficient regarding the duality σ(C0(K)⁎,C0(K)). It will be displayed that the value of this stability coefficient strongly depends on the distribution of the accumulation points of K. Finally, we prove that the dual-fixed point property is completely ‘‘unstable” for the Banach-Mazur distance in most of nonreflexive Banach spaces.