A renorming in some Banach spaces with applications to fixed point theory

Abstract

We consider a Banach space X endowed with a linear topology τ and a family of seminorms { R k ( ⋅ ) } which satisfy some special conditions. We define an equivalent norm ⦀ ⋅ ⦀ on X such that if C is a convex bounded closed subset of ( X , ⦀ ⋅ ⦀ ) which is τ-relatively sequentially compact, then every nonexpansive mapping T : C → C has a fixed point. As a consequence, we prove that, if G is a separable compact group, its Fourier–Stieltjes algebra B ( G ) can be renormed to satisfy the FPP. In case that G = T , we recover P.K. Lin's renorming in the sequence space ℓ 1 . Moreover, we give new norms in ℓ 1 with the FPP, we find new classes of nonreflexive Banach spaces with the FPP and we give a sufficient condition so that a nonreflexive subspace of L 1 ( μ ) can be renormed to have the FPP.