The almost fixed point property is not invariant under isometric renormings

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Observe that every reflexive Banach space has set-stability of the AFPP under isometric renormings.
Bearing in mind Definition 4 and the last conclusion in Theorem 1, we can ask if any Banach space has set-stability of the AFPP under isometric renormings.
The purpose of the present note is to give a negative answer to this question by showing a wide class of spaces which have not set-stability of the AFPP under isometric renormings.

Preliminaries
In [19] I. Shafrir introduced the following concept: Definition 5 A sequence (x n ) in a Banach space X is called a directional sequence if: (i) x n → ∞ (ii) There is b ≥ 0 such that for all n 1 < n 2 < · · · < n l He called a closed convex set C ⊂ X directionally bounded if it contains no directional sequences and proved the following criterium to determine when C verifies this property.

Theorem 2 A closed convex set C in a Banach space X is directionally bounded if for every sequence {x n } in C such that x n → ∞ and for every f in the unitary ball of X
In the same work he showed that the condition of directionally boundedness is equivalent to the AFPP in Banach spaces.

Theorem 3 A closed convex set C in a Banach space X has the AFPP if and only if it is directionally bounded.
Using some of the tools developed by Shafrir in his study of the AFPP, in [9] the authors proved the following characterization for reflexive spaces in terms of the AFPP.
The space of sequences of summable modulus and the space of sequences converging to 0 are denoted respectively by 1 and c 0 . If we write 1 or c 0 it is understood that 1 The following results give conditions under which a (isomorphic) copy of c 0 or 1 is complemented in a Banach space X . We include them in order to exemplify some applications of our main result. The first result is an immediate consequence of Proposition 1.8 in [14].
Theorem 5 Let X be a Banach space X with an unconditional basis. Then every copy of 1 in X contains a complemented copy of 1 (complemented in X ). Throughout this work we consider only real Banach spaces. We denote by P(X ) the collection of equivalent norms to a fixed norm of a Banach space X .

Non stability of the AFPP under Banach-Mazur distance 1
We start proving our main result for the particular cases of 1 and c 0 .
has not set-stability of the AFPP under isometric renormings.
Proof Let (e n ) denote the canonical basis of X and let W = [e 2n−1 ], where [e 2n−1 ] denotes the closed linear span of {e 2n−1 : n ∈ N}. By Theorem 4 there is an equivalent norm | · | ∈ P(W ) such that the collections of directionally bounded sets in (W , · ) and (W , | · |) differ, so without loss of generality we may assume that there is a closed convex unbounded set C ⊂ W which is directionally bounded with respect to the norm · but it is not with respect to the norm | · |. Let P : X → W be such that if x = ∞ n=1 a n e n ∈ X , then Px = ∞ n=1 a 2n−1 e 2n−1 . Define · 1 and · 2 ∈ P(X ) such that Since the inclusions i 1 : (W , · ) → (X , · 1 ) and i 2 : (W , | · |) → (X , · 2 ) are isometries on their images, C has the AFPP with respect to the norm · 1 , but it has not the AFPP for the norm · 2 . Observe that if x = ∞ n=1 a n e n ∈ X , the function T : is an onto linear isometry.
We can generalize the conclusion in Lemma 1 to the class of spaces containing a complemented subspace isomorphic to c 0 or 1 .

Theorem 9
Let (X , · ) be a Banach space containing a complemented isomorphic copy of c 0 or 1 . Then there is · 1 ∈ P(X ) such that (X , · 1 ) has not set-stability of the AFPP under isometric renormings.
Proof Let V be a complemented subspace of X isomorphic to Y , where Y is c 0 or 1 and denote by | · | 1 , | · | 2 the two norms on Y , garanteed by Lemma 1, such that the families of sets be isomorphisms. If we consider on V the norms | · | 1 and | · | 2 ∈ P(V ) such that be the isometry considered in Lemma 1. Since T is an onto linear isometry, we have that L : (V , | · | 1 ) → (V , | · | 2 ) defined as L = R −1 2 T R 1 is an onto linear isometry. Let P : X → V be a linear bounded projection from X onto V and define · 1 , · 2 ∈ P(X ) such that If S : (X , · 1 ) → (X , · 2 ) is defined as Sx = (I − P)x + L Px it is easy to check that S is an isomorphism and furthermore Below we show a wide variety of cases in which we can apply Theorem 9.

Lemma 2 Let X be a non reflexive Banach space that satisfies any of the following conditions:
(i) X has an unconditional Schauder Basis.
(ii) X contains an isomorphic copy of c 0 and X does not contain an isomorphic copy of 1 .
(iii) X is separable and contains an isomorphic copy of c 0 .
(iv) X * contains an isomorphic copy of c 0 .
(v) X has an uncountable unconditional Schauder basis [11] and contains an isomorphic copy of 1 .
Then there is a norm · 1 ∈ P(X ) such that (X , · 1 ) has not set-stability of the AFPP under isometric renormings.
Proof (i) By James theorem ([16, Corollary 4.4.23]) X contains a subspace Y isomorphic to c 0 or 1 . If Y is isomorphic to c 0 , Theorem 8 implies that Y is complemented. If Y is isomorphic to 1 , by Theorem 5 we can assume that Y is complemented. So in either case, the hypotheses of Theorem 9 are satisfied. Similarly we prove (ii)-(v) using Theorem 6, Theorem 8, Theorem 7 and Theorem 1a in [11] respectively. Lemma 2 offers a large class of examples of Banach spaces in which we can conclude the non set-stability of the AFPP under isometric renormings. However, it is still possible that given a Banach space X , we can find two non-isometric norms · 1 , · 2 in P(X ), such that (X , · 1 ) and (X , · 2 ) share the collection of sets with the AFPP and d((X , · 1 ), (X , · 2 )) = 1. In the following, we illustrate this situation in the particular case of X = c 0 .

Lemma 3 Consider the spaces X
It is easy to see that T n is an isomorphism such that T n → 1 and T −1 n → 1.
We want to compare the collections of directionally bounded sets of the spaces X and Y defined in Lemma 3. In order to do this, by Theorem 2, it is very useful to know X * and Y * . The space X * was described in [4], so we proceed to determine Y * . Straightforward calculations allow us to prove the following result: Then the set of extreme points with non-negative coordinates in the unitary ball of Y N denoted as ξ + (Y N ) satisfies: Where (e n ) is the canonical basis of c 0 .
From the last proposition it follows that: The next proposition gives a criterium to distinguish between directional and non directional sequences in the space (c 0 , | · |). Bearing in mind Proposition 2, the proof of this result is analogous to the proof of proposition 9 in [4].
Proposition 3 Let C be a closed convex unbounded set in c 0 . Suppose that C is directionally bounded in (c 0 , | · |). Let (x n ) ⊂ C be a sequence such that lim n→∞ x n ∞ = ∞, with x n = (x n (k)) ∞ k=1 . Then for every n 0 , k 0 ∈ N there exist n > n 0 and k > k 0 such that x n ∞ = |x n (k)|.
The following result describes a sufficient condition in c 0 for an unbounded sequence not to be directional. [4]) Let (x n ) ⊂ c 0 be a sequence such that lim n→∞ x n ∞ = ∞. If for every n 0 , k 0 ∈ N there exists n > n 0 and k > k 0 such that x n ∞ = |x n (k)|, then (x n ) is not a directional sequence in (c 0 , · ∞ ).

Proposition 4 (Proposition 10 in
Using Propositions 3 and 4 we can establish a relationship between the collections of directionally bounded sets in (c 0 , | · |) and c 0 .
Theorem 10 Let C be a convex, closed and unbounded subset of c 0 . Then C is directionally bounded in (c 0 , · ∞ ) if and only if it is directionally bounded in (c 0 , | · |).
Proof Let C ⊂ (c 0 , | · |) be a closed convex unbounded and directionally bounded set and (x n ) ⊂ C a sequence such that | x n | → ∞. By Proposition 3 for every n 0 , k 0 ∈ N there are n > n 0 and k > k 0 such that x n ∞ = |x n (k)| and Proposition 4 implies that (x n ) is not a directional sequence in (c 0 , · ∞ ). Conversely, suppose that C ⊂ (c 0 , · ∞ ) is a closed convex and unbounded set. If (x n ) is a directional sequence in (c 0 , | · |). Take b ≥ 0 as given by the definition of directional sequence. Let n 1 < n 2 < · · · < n s . Then By the triangle inequality: From this: Finally, from the above theorem, we conclude that there exist two equivalent renormings of a Banach space X whose Banach-Mazur distance is 1 and they share the collection of sets with the AFPP. Proof Theorem 11 in [4] implies that C is directionally bounded in (c 0 , · ) if and only if C is directionally bounded in c 0 . The conclusion follows from Theorems 3 and 10. The last statement was proved in Lemma 3.
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