The Fixed Point Property via Dual Space Properties

A Banach space has the weak fixed point property if its dual space has a weak∗ sequentially compact unit ball and the dual space satisfies the weak∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε > 0 such that, for every infinite subset A of the unit sphere of the dual space, A ∪ (−A) fails to be (2 − ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property. Determining conditions on a Banach space X so that every nonexpansive mapping from a nonempty, closed, bounded, convex subset of X into itself has a fixed point has been of considerable interest for many years. A Banach space has the fixed point property if, for each nonempty, closed, bounded, convex subset C of X, every nonexpansive mapping of C into itself has a fixed point. A Banach space is said to have the weak fixed point property if the class of sets C above is restricted to the set of weakly compact convex sets; and a Banach space is said to have the weak fixed point property if X is a dual space and the class of sets C is restricted to the set of weak compact convex subsets of X. A well-known open problem in Banach spaces is whether every reflexive Banach space has the fixed point property for nonexpansive mappings. The question of whether more restrictive classes of reflexive spaces, such as the class of superreflexive Banach spaces or Banach spaces isomorphic to the Hilbert space l, have the fixed point property has also long been open and has been investigated by many authors [8, 14, 15, 17]. Recently, J. Garćıa-Falset, E. Llorens-Fuster, and E.M. MazcuñanNavarro [7] proved that uniformly nonsquare Banach spaces, a sub-class of the superreflexive spaces, have the fixed point property. In this article, it is shown that the larger class of E-convex Banach spaces have the fixed point property. The E-convex Banach spaces, introduced by S.V.R. Naidu and K.P.R. Sastry [18], are a class of Banach spaces lying strictly between the uniformly nonsquare Banach spaces and the superreflexive spaces (see also [1]). The second geometric property of Banach spaces that is considered in this article is the weak uniform Kadec-Klee property in a dual Banach space. A dual space X has the weak uniform Kadec-Klee property if, for every ε > 0, there exists δ > 0 such that, if (xn) is a sequence in the unit ball of X ∗ converging weak to x and the separation constant sep(xn) def = inf{‖xn − x ∗ m‖ : m 6= n} > ε, then ‖x ‖ < 1− δ. It is Date: April 4, 2008. 1 2 P.N. DOWLING, B. RANDRIANANTOANINA, AND B. TURETT well-known [6] that, if X has weak uniform Kadec-Klee property, then X has the weak fixed point property. If, in addition, the unit ball of X is weak sequentially compact, more is true: Theorem 3 notes that, if X has weak uniform Kadec-Klee property and the unit ball of X is weak sequentially compact, then X has the weak fixed point property. As a consequence of Theorem 3, it is noted that several nonreflexive Banach spaces such as quotients of c0 and C(T )/A0, the predual of H , have the weak fixed point property. Since the proofs of the main theorems in this paper will require elements of the proof that uniformly nonsquare Banach spaces have the fixed point property, a complete proof of this known result is presented. The proof presented here is a distillation of the original proof and combines elements from [5, Th. 2.2] and [7, Th. 3.3]. Recall that a Banach space X is uniformly nonsquare [10] if there exists δ > 0 such that, if x and y are in the unit ball of X, then either ‖(x+y)/2‖ < 1−δ or ‖(x−y)/2‖ < 1−δ. The general set-up in proving that a Banach space has the weak fixed point property has, by now, become standard fare. If a Banach space X fails to have the weak fixed point property, there exists a nonempty, weakly compact, convex set C in X and a nonexpansive mapping T : C → C without a fixed point. Since C is weakly compact, it is possible by Zorn’s Lemma to find a minimal T -invariant, weakly compact, convex subset K of C such that T has no fixed point in K. Since the diameter of K is positive (otherwise K would be a singleton and T would have a fixed point in K), it can be assumed that the diameter of K is 1. It is well-known that there exists an approximate fixed point sequence (xn) for T in K and, without loss of generality, we may assume that (xn) converges weakly to 0. For details on this general set-up, see [8, Chapter 3]. Theorem 1 ( Garćıa-Falset, Llorens-Fuster, and E.M. Mazcuñan-Navarro [7]). Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. Proof. Assume that a Banach space X fails to have the fixed point property. Since uniformly nonsquare spaces are reflexive [10], the fixed point property and the weak fixed point property coincide for X. Therefore there exists a nonexpansive map T : K → K without a fixed point where K is a minimal T -invariant set in X with diameter 1. Let (xn) be an approximate fixed point sequence for T in K and assume that (xn) converges weakly to 0. Consider the set in l(X)/c0(X) defined by [W ] = { [zn] ∈ [K] : ‖[zn]− [xn]‖ ≤ 1/2 and lim sup n lim sup m ‖zm − zn‖ ≤ 1/2 } . It is easy to check that [W ] is closed, bounded, convex, nonempty (since [ 2 xn] is in the set), and [T ]-invariant where [T ][zn] def = [T (zn)]. So, by a result of Lin [13], sup [zn]∈[W ] ‖[zn]− x‖ = 1 for each x ∈ K. In particular, with x = 0, sup [zn]∈[W ] ‖[zn]‖ = 1 . THE FIXED POINT PROPERTY VIA DUAL SPACE PROPERTIES 3 Let ε > 0 and choose [zn] ∈ [W ] with ‖[zn]‖ > 1 − ε. Let (yj) = (znj ) be a subsequence of (zn) such that lim ‖yn‖ = ‖[zn]‖ and (yn) converges weakly to an element y in K. There is no loss in generality in assuming that ‖yn‖ > 1 − ε for all n ∈ N and in choosing y n ∈ X ∗ so that ‖y n‖ = 1, y ∗ n(yn) = ‖yn‖, and (y n) converges weak ∗ to y. (This is possible because the fixed point property is separably-determined [8, page 35]; so there is no loss in generality in assuming that BX∗ is weak -sequentially compact.) From the definition of [W ] and the weak lower semicontinuity of the norm, it follows that, if n is large enough, ‖yn − y‖ ≤ lim inf m ‖yn − ym‖ < 1 + ε 2 and ‖y‖ ≤ lim inf j ‖yj − xnj‖ < 1 + ε 2 . Therefore, with un = 2 1+ε (yn − y) and u = 2 1+ε y, ‖un + u‖ = ∥∥∥∥ 2 1 + ε (yn − y) + 2 1 + ε y ∥∥∥∥ = 2 1 + ε ‖yn‖ > 2 1− ε 1 + ε > 2(1− 2ε) if n ∈ N is large enough. Applying the weak lower semicontinuity of the norm again, it follows that lim inf m ‖(un − um) + u‖ ≥ ‖un + u‖ > 2(1− 2ε) if n ∈ N is large enough. So, by taking another subsequence if necessary, we can assume that ‖un + u‖ > 2(1− 2ε) and ‖(un − um) + u‖ > 2(1− 2ε) for all n and all m > n. Furthermore, since y m w∗ → y, lim inf m ‖(un − um)− u‖ = lim inf m ‖(um + u)− un‖ ≥ lim inf m y m ( (um + u)− un ) = lim inf m ( ‖um + u‖ − y ∗ m(un) ) ≥ 2(1− 2ε)− y(un) . Then, since un w → 0, it follows that lim infm ‖(un − um)− u‖ > 2(1− 3ε) if n is large enough. Therefore, for n large enough and m > n also large enough, both ‖(un − um) + u‖ > 2(1− 3ε) and ‖(un − um)− u‖ > 2(1− 3ε) hold. Since ε > 0 is arbitrary, ‖un − um‖ < 1, and ‖u‖ < 1, the above inequalities imply that X fails to be uniformly nonsquare, a contradiction which finishes the proof. We want to refine the sequences (xnj ), (yj), and (y ∗ j ) that appear in the proof of Theorem 1. Recall the result of Goebel and Karlovitz [8, page 124]: If K is a minimal T -invariant, weakly compact, convex set for the nonexpansive map T and (xn) is an 4 P.N. DOWLING, B. RANDRIANANTOANINA, AND B. TURETT approximate fixed point sequence for T in K, then the sequence (‖xn−x‖) converges to the diameter of K for every x in K. Fix ε > 0 and set x̃1 = xn1 , ỹ1 = y1, and ỹ ∗ 1 = y ∗ 1. Then, by the Goebel-Karlovitz Lemma, there exists j1 > 1 such that min{‖x̃1 − xnj1‖, ‖ỹ1 − xnj1‖} > 1 − ε. Set x̃2 = xnj1 , ỹ2 = yj1, and ỹ ∗ 2 = y ∗ j1 . Another application of the Goebel-Karlovitz Lemma yields j2 > j1 such that min i=1,2 {‖x̃i − xnj2‖, ‖ỹi − xnj2‖} > 1− ε. Set x̃3 = xnj2 , ỹ3 = yj2, and ỹ ∗ 3 = y ∗ j2 . Continuing in this manner, we obtain sequences (x̃n) and (ỹn) inK (and BX) and a sequence (ỹ ∗ n ) in SX∗ satisfying min i 1−ε for all n ∈ N and ỹ ∗ n (ỹn) = ‖ỹn‖ > 1−ε for all n ∈ N . In the following, these sequences are renamed by omitting the tildes. The following result is a summary of several easy computations. Lemma 2. Let X be a Banach space whose dual unit ball is weak ∗ sequentially compact and assume that X fails the weak fixed point property. Given ε > 0, there exist sequences (yn) in BX and (y ∗ n) in SX∗ and elements y ∈ BX and y ∗ ∈ BX∗ satisfying: (1) yn w → y and y n w ∗ → y; (2) For every n ∈ N, 1− ε < ‖yn‖ = y ∗ n(yn) ≤ 1; (3) For every n ∈ N, 1−3ε 2 < y n(y) ≤ ‖y‖ < 1+ε 2 ; (4) 1−3ε 2 < ‖yn − y‖ < 1+ε 2 ; (5) If n 6= m, then 1−3ε 2 < ‖yn − ym‖ < 1+ε 2 ; (6) If n 6= m, then 1−3ε 2 < y n(ym) < 1+2ε 2 ; (7) 1−3ε 2 ≤ y(y) ≤ 1+ε 2 ; Proof. Claims (1) and (2), the third inequality in (3), and the second inequalities in (4) and (5) are immediate from the proof of Theorem 1. Then ‖y‖ ≥ y n(y) = y ∗ n(yn)− y ∗ n(yn − y) > (1− ε)− 1 + ε