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

We consider a Banach space X endowed with a linear topology τ and a family of seminorms {Rk(·)} 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 1. Moreover, we give new norms in 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 L1(μ) can be renormed to have the FPP. © 2009 Elsevier Inc. All rights reserved.