Renorming of and the fixed point property

For any k ∈ N , let P k denote the natural projections on ℓ 1 . Let | | | ⋅ | | | be an equivalent norm of ℓ 1 that satisfies all of the following four conditions:(1) are α > 4 and a positive (decreasing) sequence ( α n ) in ( 0 , 1 ) such that for any normalized block basis { f n } of ( ℓ 1 , | | | ⋅ | | | ) and x ∈ ℓ 1 with P k − 1 ( x ) = x and | | | x | | | < α k , lim sup n → ∞ | | | f n + x | | | ⩽ 1 + | | | x | | | α . are two strictly decreasing sequences { β k } and { γ k } with lim k → ∞ β k = 0 and lim k → ∞ γ k = 1 such that for any normalized block basis { f n } of ( ℓ 1 , | | | ⋅ | | | ) and x with ( I − P k ) ( x ) = x , lim inf n → ∞ | | | f n + x | | | ⩾ 1 − β k + γ k − 1 | | | x | | | . y k ∈ N , ‖ I − P k ‖ = 1 . it ball of ( ℓ 1 , | | | ⋅ | | | ) is σ ( ℓ 1 , c 0 ) -closed. is article, we prove that the space ( ℓ 1 , | | | ⋅ | | | ) has the fixed point property for the nonexpansive mapping. This improves a previous result of the author.