B. Turett

The optimality of James’s distortion theorems

A renorming of `1, explored here in detail, shows that the copies of `1 produced in the proof of the Kadec-Pe lczynski theorem inside nonreflexive subspaces of L1[0, 1] cannot be produced inside general nonreflexive spaces that contain copies of `1. Put differently, James’s distortion theorem producing oneplus-epsilon-isomorphic copies of `1 inside any isomorphic copy of `1 is, in a certain sense, optimal. A similar renorming of c0 shows that James’s distortion theorem for c0 is likewise optimal. James’s distortion theorems for `1, the space of absolutely summable sequences of scalars, and c0, the space of null sequences of scalars, are well-known [J]. The former states that, whenever a Banach space contains a subspace isomorphic to `1, the Banach space contains subspaces that are almost isometric to `1. Several of the authors of this article, individually and in concert, have tried to use this feature of `1 to determine if all (equivalent) renormings of `1 fail to have the fixed point property for nonexpansive mappings (the FPP); i.e. if, in any renorming of `1, there exist a nonempty, closed, bounded and convex subset C and a nonexpansive self-map T of C without a fixed point. The basis of these attempts was to use the fact that `1 in its usual norm fails to have the fixed point property and, since each renorming of `1 contains subspaces almost isometric to `1, a perturbation of the usual example would hopefully produce a nonexpansive self-map of a nonempty, closed, bounded, convex set in any renorming of `1. Similar attempts in c0 were also made. What appeared to be needed in these attempts were strengthened versions of James’s distortion theorems. To be specific, James’s theorem for `1 states that if a Banach space X with norm ‖ · ‖ contains an isomorphic copy of `1, then, for each > 0, there exists a sequence (xk) in the unit sphere of X such that (1− ) ∑∞ k=1 |tk| ≤ ‖ ∑∞ k=1 tkxk‖ ≤ ∑∞ k=1 |tk| for all (tk) ∈ `1. The proof of the theorem shows even more than the statement indicates. The sequence (xk) may be chosen to have the additional property that, if ( n) is a sequence of positive numbers decreasing to 0, then for each n, (1 − n) ∑∞ k=n |tk| ≤ ‖ ∑∞ k=n tkxk‖ ≤ ∑∞ k=n |tk| , for all (tk) ∈ `1. That is, for each δ > 0, by ignoring a finite number of terms at the beginning of the sequence (xk), one obtains copies of `1 which are (1 + δ)-isomorphic to `1. This Received by the editors May 8, 1995 and, in revised form, July 7, 1995. 1991 Mathematics Subject Classification. Primary 46B03, 46B20.

Weak compactness is equivalent to the fixed point property in c~0

A nonempty, closed, bounded, convex subset of c 0 has the fixed point property if and only if it is weakly compact.

The fixed point property for subsets of some classical Banach spaces

We prove a general result about the stability of the fixed point property in closed bounded convex subsets of certain Banach spaces. This allows us to characterize those closed bounded convex subsets of L1[0, 1] that are weakly compact. Similarly, we characterize the closed bounded convex subsets of !1 that are compact. Generalizations of these results to the setting of noncommutative L1-spaces are also obtained.

SOME EXAMPLES CONCERNING NORMAL AND UNIFORM NORMAL STRUCTURE IN BANACH SPACES

(Received 23 February 1989)Communicated by S. YamamuroAbstractExamples are given that show the following: (1) normal structure need not be inherited byquotient spaces; (2) uniform normal structure is not a self-dual property; and (3) no degree ofA>uniform rotundity need be present in a space with uniform normal structure.1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 46 B 20.

Remarks on James's distortion theorems

If a Banach space X contains a complemented subspace isomorphic to l 1 and if e > 0, then there exists a subspace Y of X and a projection P from X onto Y such that Y is (1 + e)-isometric to l 1 and ∥ P ∥ ≤ 1 + e. A stronger result for c 0 is proved for Banach spaces whose dual unit ball is weak sequentially compact.

Some fixed point results in l 1 and c 0

When James [10] proved that neither l1 nor c0 is distortable, he provided a tool which appeared to be useful in considering the question of whether l1 or c0 could be renormed to have the !xed point property. The gist of the desired proof was that, since both spaces admit !xed point-free isometries on closed bounded convex sets and all renormings of l1 or c0 contain almost isometric copies of l1 or c0, then perturbations of the isometries would hopefully produce nonexpansive self-maps of closed bounded convex subsets without !xed 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 !xed point property as noted in [9]), connections between spaces containing good copies of l1 or c0 and the failure of the !xed point property have been investigated in several articles [4–6]. In this article, the authors continue to investigate James’s distortion theorems and their relationship to !xed points and to the more restrictive renormings of l1 and c0 considered in the above articles. Simply stated, James’s distortion theorems state that Banach spaces which contain isomorphic copies of l1 (respectively, c0) contain almost isometric copies of l1 (respectively, c0). In fact, the proofs that James [10] gives for these theorems shows a bit more.