Renormings and fixed point property in non-commutative L1L1-spaces II: Affine mappings

In this paper, we prove that every noncommutative L1L1-space associated to a finite von Neumann algebra can be renormed to satisfy the fixed point property for nonexpansive affine mappings. Particular examples are L1(R)L1(R), where RR is the hyperfinite II1II1 factor and the function spaces L1[0,1]L1[0,1] and L1(μ)L1(μ) for any σσ-finite measure space. This property does not hold for the usual ‖⋅‖1‖⋅‖1 norm.