The fixed point property under renorming in some classes of Banach spaces

Abstract

Assume that Y is a Banach space such that R ( Y ) < 2 where R ( ⋅ ) is García-Falset’s coefficient, and X is a Banach space which can be continuously embedded in Y . We prove that X can be renormed to satisfy the weak Fixed Point Property (w-FPP). On the other hand, assume that K is a scattered compact topological space such that K ( ω ) = 0̸ and C ( K ) is the space of all real continuous functions defined on K with the supremum norm. We will show that C ( K ) can be renormed to satisfy R ( C ( K ) ) < 2 . Thus, both results together imply that any Banach space which can be continuously embedded in C ( K ) , K as above, can be renormed to satisfy the w-FPP. These results extend a previous one about the w-FPP under renorming for Banach spaces which can be continuously embedded in c 0 ( Γ ) . Furthermore, we consider a metric in the space P of all norms in C ( K ) which are equivalent to the supremum norm and we show that for almost all norms in P (in the sense of porosity) C ( K ) satisfies the w-FPP.