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

• Published in: Journal of functional analysis 2010-05, Vol.258 (10), p.3452-3468
• Main Authors: Linares, Carlos A. Hernandez, Japon, Maria A.
• Format: Article
• Language: English
• Subjects: Fixed point theory; Fourier algebras; Fourier–Stieltjes algebra; Nonexpansive mappings; Renorming theory; Topology of convergence locally in measure
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2009.10.025

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.