Direct sums of renormings of ℓ1ℓ1 and the fixed point property

Let γnγn be an increasing sequence in (0,1)(0,1) that converges to 1, and let ⦀⋅⦀⦀⋅⦀ be the equivalent norm of ℓ1ℓ1 defined by ⦀(ak)⦀=supn∈Nγn∑k=n∞|ak|. In this article, we show that for any m>1m>1, the space (∑i=1m⊕(ℓ1,⦀⋅⦀))1 is not isometrically isomorphic to any subspace of (ℓ1,⦀⋅⦀)(ℓ1,⦀⋅⦀) and it has the fixed point property.