Separable Lindenstrauss spaces whose duals lack the weak$^*$ fixed point property for nonexpansive mappings

Abstract

Abstract. In this paper we study the w * -fixed point property for nonexpansive mappings. First we show that the dual space X * lacks the w * -fixed point property whenever X contains an isometric copy of the space c. Then, the main result of our paper provides several characterizations of weak-star topologies that fail the fixed point property for nonexpansive mappings in ℓ 1 space. This result allows us to obtain a characterization of all separable Lindenstrauss spaces X inducing the failure of w * -fixed point property in X * .

Stability constants of the weak⁎ fixed point property for the space ℓ1

“…Then, clearly lim n→∞ |α − α n | ℓ1 = 0. Moreover, by Proposition 2.1 in [5], the space W αn contains an isometric copy of c, for every n. By Lemma 2.1, we have that lim n→∞ d(W α , W αn ) = 1, which completes the proof.…”

“…For a detailed study of this class of spaces we refer the reader to [4] and [5]. Here we recall only that W * α = ℓ 1 .…”

“…Proof. By Theorem 4.1 in [5] there exists a quotient X/Y of X that is isometric to a space W α with |α| ℓ1 = 1. Let us first suppose in addition that W α contains a subspace isometric to c. Now, let us consider the onto map: πT −1 : c 0 → W α , where π : X → X/Y is the quotient map.…”

“…It is easy to check that φ is an onto isomorphism. Moreover, if α ∈ S ℓ1 , then it holds φ φ −1 = 3 and the space W α contains an isometric copy of c (see Proposition 2.1 in [5]). Hence, Theorem 2.1 in [11] implies that ϕ ϕ −1 ≥ 3 for every isomorphism ϕ from W α onto c 0 .…”

“…For instance, this situation occurs when we consider the space ℓ 1 and its preduals c 0 and c where it is well-known (see [12]) that ℓ 1 has the σ(ℓ 1 , c 0 )-fpp whereas it lacks the σ(ℓ 1 , c)-fpp. Necessary and sufficient conditions for a predual X of ℓ 1 to be a space such that the σ(ℓ 1 , X)-fpp holds are proved in [5]. The present paper concerns the investigation of the stability of the σ(ℓ 1 , X)-fpp carrying on the study developed in [7].…”