Title

Renorming of ℓ1 and the fixed point property

Abstract

For any k ∈ N, let Pk denote the natural projections on ℓ1. Let | | | ṡ | | | be an equivalent norm of ℓ1 that satisfies all of the following four conditions: (1)There are α > 4 and a positive (decreasing) sequence (αn) in (0, 1) such that for any normalized block basis {fn} of (ℓ1, | | | ṡ | | |) and x ∈ ℓ1 with Pk - 1 (x) = x and | | | x | | | < αk,under(lim sup, n → ∞) | | | fn + x | | | ≤ 1 + frac(| | | x | | |, α) .(2)There are two strictly decreasing sequences {βk} and {γk} withunder(lim, k → ∞) βk = 0 and under(lim, k → ∞) γk = 1 such that for any normalized block basis {fn} of (ℓ1, | | | ṡ | | |) and x with (I - Pk) (x) = x,under(lim inf, n → ∞) | | | fn + x | | | ≥ 1 - βk + γk- 1 | | | x | | | .(3)For any k ∈ N, {norm of matrix} I - Pk {norm of matrix} = 1.(4)The unit ball of (ℓ1, | | | ṡ | | |) is σ (ℓ1, c0)-closed. In this article, we prove that the space (ℓ1, | | | ṡ | | |) has the fixed point property for the nonexpansive mapping. This improves a previous result of the author. © 2009 Elsevier Inc. All rights reserved.

Publication Title

Journal of Mathematical Analysis and Applications

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