Renorming of ℓ1ℓ1 and the fixed point property

For any k∈Nk∈N, let PkPk denote the natural projections on ℓ1ℓ1. Let |||⋅||||||⋅||| be an equivalent norm of ℓ1ℓ1 that satisfies all of the following four conditions:(1)There are α>4α>4 and a positive (decreasing) sequence (αn)(αn) in (0,1)(0,1) such that for any normalized block basis {fn}{fn} of (ℓ1,|||⋅|||)(ℓ1,|||⋅|||) and x∈ℓ1x∈ℓ1 with Pk−1(x)=xPk−1(x)=x and |||x|||<αk|||x|||<αk,lim supn→∞|||fn+x|||⩽1+|||x|||α.(2)There are two strictly decreasing sequences {βk}{βk} and {γk}{γk} withlimk→∞βk=0andlimk→∞γk=1 such that for any normalized block basis {fn}{fn} of (ℓ1,|||⋅|||)(ℓ1,|||⋅|||) and x   with (I−Pk)(x)=x(I−Pk)(x)=x,lim infn→∞|||fn+x|||⩾1−βk+γk−1|||x|||.(3)For any k∈Nk∈N, ‖I−Pk‖=1‖I−Pk‖=1.(4)The unit ball of (ℓ1,|||⋅|||)(ℓ1,|||⋅|||) is σ(ℓ1,c0)σ(ℓ1,c0)-closed. In this article, we prove that the space (ℓ1,|||⋅|||)(ℓ1,|||⋅|||) has the fixed point property for the nonexpansive mapping. This improves a previous result of the author.