The τ-fixed point property for left reversible semigroups

Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 DOI 10.1186/s13663-015-0357-7 RESEARCH Open Access The τ -fixed point property for left reversible semigroups Francisco E Castillo-Santos1 and Maria A Japón2* * Correspondence: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Tarfia s/n, Sevilla, 41012, Spain Full list of author information is available at the end of the article 2 Abstract In this article we use the generalized Gossez-Lami Dozo property and the Opial condition to study the fixed point property for left reversible semigroups in separable Banach spaces. As a consequence, some previous results will be deduced and new examples of Banach spaces satisfying the fixed point property for left reversible semigroups are shown. We will also extend some previous theorems when we consider the semigroup formed by a unique nonexpansive mapping and its iterates. MSC: 46B03; 47H09; 47H10 Keywords: Schauder basis; sequentially separating norms; fixed point property; nonexpansive mappings; renorming theory; Schur property 1 Introduction A semigroup S is said to be a semitopological semigroup if S is equipped with a Hausdorff topology such that for each a ∈ S, the two mappings from S into S defined by s → as and s → sa are continuous. A semitopological semigroup S is said to be left reversible if any two nonempty closed right ideals of S have nonempty intersection. Clearly every Abelian semitopological semigroup and every semitopological group are left reversible. Also left amenable and in particular amenable semitopological semigroups are left reversible []. Let C be a subset of a Banach space X and let S be a semitopological semigroup. A nonexpansive action of S on the set C is a map φ : S × C → C, denoted by φ(s, u) = s(u) (or su), which satisfies: (i) ts(u) = t(su) for all t, s ∈ S and u ∈ C. (ii) For all u ∈ C, the function s ∈ S → s(u ) ∈ C is continuous. (iii) For every s ∈ S, the mapping u ∈ C → s(u) ∈ C is nonexpansive. A subset C is said to verify the fixed point property for left reversible semigroups if for every left reversible semitopological semigroup S and for every nonexpansive action φ : S × C → C, the set Fix(S) := {u ∈ C : t(u) = u, ∀t ∈ S} is nonempty. Definition . Let X be a Banach space and τ be a topology on X. It is said that X has the τ fixed point property (τ -FPP) for left reversible semigroups if every closed, convex, bounded subset C which is τ -compact has the fixed point property for left reversible semigroups. Given a nonexpansive mapping T, if we replace the left reversible semigroup by the discrete and Abelian semigroup {T, T  , T  , . . .} acting from C to C, Definition . becomes the © 2015 Castillo-Santos and Japón. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 2 of 19 usual definition of the τ -FPP for nonexpansive mappings. There exist some Banach spaces failing the w-FPP [] and therefore they fail the w-FPP for left reversible semigroups (we can consider the semigroup S = {T, T  , T  , . . .} where T is the fixed point free nonexpansive mapping in the well-known Alspach example []). In  Kirk proved that every Banach space with weak normal structure satisfies the w-FPP for nonexpansive mappings. In a similar way it can be proved that weak∗ normal structure implies the weak∗ -FPP in dual Banach spaces. In the seventies Kirk’s result was generalized by Lim [], Holmes and Lau [] in the setting of nonexpansive actions of left reversible semigroups, that is, weak normal structure implies the w-FPP for left reversible semigroups. In the case of dual Banach spaces, such a general statement is still unknown for the weak∗ normal structure and the weak∗ -fixed point property for left reversible semigroups (see Open Problem . in []). Particular examples of dual Banach spaces are known to satisfy the weak∗ -FPP for left reversible semigroups. In  Lim [] proved that the sequence space  satisfies the weak∗ -FPP for left reversible semigroups. In , Lau and Mah in [] generalized Lim’s result by proving that the Fourier-Stieltjes algebra B(G) of a separable compact group verifies the weak∗ -FPP for left reversible semigroups. Notice that if G is the torus group, then B(G) is isometric to  (Z). In , Randrianantoanina [] proved that the space T (H) of trace class operators on a Hilbert space also satisfies the weak∗ -FPP for left reversible semigroups. He also proved the same property for the Hardy Banach space []. However, the techniques used in the previous articles cannot be extended to more general dual Banach spaces since they are mainly based on the following fact: in the abovementioned Banach spaces, the asymptotic center of a weak∗ compact set with respect to a decreasing net of bounded subsets is proved to be either norm compact or weakly compact. This is not true for every weak∗ compact set in a dual Banach space, as we will later check in Example .. In , Randrianantoanina [] proved that the Banach space L [, ] or, more generally, every noncommutative L -space associated to a finite von Neumann algebra satisfies the fixed point property for left reversible semigroups with respect to the abstract measure topology τ (the convergence in measure topology in case of L [, ]). Here the asymptotic centers of τ -compact sets are norm compact. In this paper we develop new arguments to deduce whether a dual Banach space satisfies the weak∗ -FPP for left reversible semigroups. More generally, we will consider τ as any translation invariant topology on a separable Banach space X and we give sufficient conditions to assure the τ -FPP for left reversible semigroups. The strict Opial condition and the generalized Gossez-Lami Dozo property will be our main tools. Most of the previous known results will be deduced from ours, but we will also achieve new examples of Banach spaces which satisfy the τ -FPP for left reversible semigroups. Here we will consider different types of topologies. Firstly we will regard the weak∗ topology in Musielak-Orlicz sequence spaces, in some renormings of  and in some other dual Banach spaces nonisomorphic to  . We will also consider the topology of the convergence locally in measure in some function spaces, the abstract measure topology in L-embedded Banach spaces and the topology of ρ-almost everywhere convergence in modular function spaces. Moreover, we will extend some known results for nonexpansive mappings to the setting of the fixed point property for left reversible semigroups. Castillo-Santos (...truncated)