When James [10] proved that neither l 1 nor c 0 is distortable, he provided a tool which appeared to be useful in considering the question of whether l 1 or c 0 could be renormed to have the ΓΏxed point property. The gist of the desired proof was that, since both spaces admit ΓΏxed point-free isometries on closed bounded convex sets and all renormings of l 1 or c 0 contain almost isometric copies of l 1 or c 0 , then perturbations of the isometries would hopefully produce nonexpansive self-maps of closed bounded convex subsets without ΓΏxed points in the renormed spaces. Although the authors have not been able to bring this idea to fruition (not surprisingly perhaps given the lack of stability of subsets with the ΓΏxed point property as noted in [9]), connections between spaces containing good copies of l 1 or c 0 and the failure of the ΓΏxed point property have been investigated in several articles [4][5][6]. In this article, the authors continue to investigate James's distortion theorems and their relationship to ΓΏxed points and to the more restrictive renormings of l 1 and c 0 considered in the above articles.
Simply stated, James's distortion theorems state that Banach spaces which contain isomorphic copies of l 1 (respectively, c 0 ) contain almost isometric copies of l 1 (respectively, c 0 ). In fact, the proofs that James [10] gives for these theorems shows a bit more.
James's distortion theorems. A Banach space X contains an isomorphic copy of l 1 if and only if; for every null sequence ( n ) in (0; 1), there exists a sequence (x n ) n in * Corresponding author.