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-Santos1and Maria A Japón2* *Correspondence: [email protected] 2Departamento 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 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 AsemigroupSissaidtobeasemitopologicalsemigroupifSisequippedwithaHausdorff topology such that for each a∈S, the two mappings from Sinto Sdefined by s→as and s→sa are continuous. A semitopological semigroup Sis said to be left reversible if any two nonempty closed right ideals of Shave 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 CbeasubsetofaBanachspaceXandletSbe a semitopological semigroup.Anonexpansiveactionof SonthesetCisamapφ:S×C→C,denotedbyφ(s,u)=s(u)(orsu), which satisfies: (i) ts(u)=t(su)for all t,s∈Sand u∈C. (ii) For all u∈C,thefunctions∈S→s(u)∈Cis continuous. (iii) For every s∈S, the mapping u∈C→s(u)∈Cis nonexpansive. AsubsetCis said to verify the fixed point property for left reversible semigroups if for every left reversible semitopological semigroup Sand for every nonexpansive action φ:S×C→C,thesetFix(S):={u∈C:t(u)=u,∀t∈S}is nonempty. Definition . Let Xbe a Banach space and τbe a topology on X.ItissaidthatXhas the τfixed point property (τ-FPP) for left reversible semigroups if every closed, convex, boundedsubsetCwhichisτ-compacthasthefixedpointpropertyforleftreversiblesemigroups. GivenanonexpansivemappingT,ifwereplacetheleftreversiblesemigroupbythediscreteandAbeliansemigroup{T,T,T,...}actingfromCtoC,Definition.becomesthe ©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 usualdefinitionoftheτ-FPPfornonexpansivemappings.ThereexistsomeBanachspaces 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 Tis 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 []inthesetting 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 resultbyprovingthattheFourier-StieltjesalgebraB(G)ofaseparablecompactgroupverifiestheweak∗-FPPforleftreversiblesemigroups.NoticethatifGisthetorusgroup,then B(G)isisometricto(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 adecreasing net of boundedsubsetsis provedtobe eithernorm compact or weakly compact.Thisisnottrueforeveryweak ∗compact set in a dual Banach space, as we will later check in Example .. In,Randrianantoanina[]provedthattheBanachspaceL[,]or,moregenerally, 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 incase of L[,]). Heretheasymptotic centers of τ-compact sets are norm compact. In this paper we develop new arguments to deduce whether a dual Banach space satisfiestheweak∗-FPPforleftreversiblesemigroups.Moregenerally,wewillconsiderτasany translation invariant topology on a separable Banach space Xand 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 knownresultswillbededucedfromours,butwewillalsoachievenewexamplesofBanach 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 nonisomorphicto.Wewillalsoconsiderthetopologyoftheconvergencelocallyinmeasure insomefunctionspaces,theabstractmeasuretopologyinL-embeddedBanachspacesand the topology of ρ-almost everywhere convergence in modular function spaces. Moreover,wewillextendsomeknownresultsfornonexpansivemappingstothesetting of the fixed point property for left reversible semigroups. Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 3 of 19 2Preliminaries We introduce some definitions and concepts. LetXbeaBanachspaceandlet{Bs}s∈AbeadecreasingnetofboundedsubsetsofX.For x∈Xand s∈A, we consider rs(x):=supx–y:y∈Bs, r(x):=infrs(x):s∈A=lim srs(x). Notice that r(·) is a continuous function for the norm topology. We defined the asymptotic radius and the asymptotic center of a set Cwith respect to the family {Bs}s∈Aas r:=infr(x):x∈C, AC{Bs}s∈A,C:= x∈C:r(x)=r. In the following example we check that asymptotic centers of weak∗-compact sets are not weakly compact in general. Example. Let Xbe the space renormed as follows: x=max n∈Nx(n)+  ∞  i=n+x(i). Since{en}isamonotonousboundedlycompleteSchauderbasisforX=(,·),thisspace is isometric to the dual of the closed subspace spanned by the orthogonal functions {e∗ n}. Thus Xisadualspaceandinfactitsweak∗topologycoincides withtheσ(,c)topology. Consider the set C:=∞  n= tnen:tn≥, ∞  n= tn≤ , which is a closed convex bounded w∗-compact set. Definetheboundedsubsets Bs={ek:k≥s},wherethe{en}aretheunitbasicvectors.In this case r(x)=limsupsx–es,anditiseasytocheckthatr(x)≥forall x∈.However, for all k,nwith k<n,  ek–en=, whichimpliesthat  enbelongstotheasymptoticcenterof Cwithrespecttothesequence {Bs}s,andthiscenterisnotweaklycompact.Therefore,theargumentsusedin[–]or[] to prove the existence of a common fixed point for left reversible semigroups of nonexpansive mappings are not useful in this example. We will later deduce that (,·)does satisfy the w∗-FPP for left reversible semigroups. Let(T,τ)beatopologicalspace.Recallthatafunction f:T→Rissaidtobeτ-sequentially lower semicontinuous (τ-slsc) if f(t)≤liminfnf(tn)foreverysequence(tn)n⊂T Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 4 of 19 withτ-limntn=t.Fromnowon,let XbeaBanachspaceandletτbe atranslationinvariant topology on X,thatis,τ-limnxn=xif and only if τ-limn(xn–x)=. Weintroducethefollowingdefinitionswhichgeneralizetwogeometricpropertieswhich are well known in case of the weak topology. These properties were attempts to get some information about the behavior of the norm on the weakly convergent sequences. The Opial condition was introduced by Opial () [] and the generalized Gossez-Lami Dozo property was introduced by Jiménez-Melado () []whenτis the weak topology. Definition. Let(X,·)beaBanachspaceandτbeatopologyonX.WewillsaythatX has thegeneralizedGossez-Lami Dozoproperty forthetopology τ(τ-GGLD)if for every (norm) bounded and τ-null sequence {xn}such that limxn=andlimn,m,n=mxn–xm exists, it is the case that limxn<lim n,m,n=mxn–xm. Definition . It is said that a Banach space (X,·) satisfies the Opial condition with respect to a topology τif liminf nxn–x<liminf nxn–x for all x∈Xwith x=x,whenever(xn)isasequenceinXwith τ-limnxn=x. The τ-GGLD property and the Opial condition will be the key to our main results. It is wellknown thatthesepropertiesarenotrelated.Wewill also illustratethisassertionwith some examples. Definition. Let(X,·)beaBanachspaceandτbeatopologyonX.ItissaidthatXis uniformlyKadec-Kleewithrespect to τ,UKK(τ),ifforevery>  there exists some δ> such that whenever {xn}nisasequenceintheclosedunitballofX,whichisτ-convergent to a point x∈Xwith infn=mxn–xm>,thenx<–δ. Associated with the UKK(τ) property, the following modulus is defined: PX,τ()=inf–x:xn≤,τ-lim nxn=x,inf n=mxn–xm>. In case that τis the weak topology the previous coefficient is known as Partington’s modulus,anditisclearthataBanachspacehastheUKK(τ)propertyifandonlyifPX,τ()> forevery∈(,].WedenotePX,τ(–)=lim→–PX,τ().Forτalineartopology,itisnot difficult to check that limxn≤–PX,τ– lim n,m,n=mxn–xm for every (norm) bounded τ-null sequence {xn}such that limn,m,n=mxn–xmexists. Therefore Xverifies the τ-GGLD property whenever Xis UKK(τ). Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 5 of 19 3 Main results Let Cbe a set and {Bs}s∈Abe a decreasing net of bounded subsets of X.Itisclearthat AC{Bs}s∈A,C= n∈Nx∈C:r(x)≤r+ n. Therefore, if the set Cis τ-sequentially compact and the function r(·)isτ-slsc, the asymptotic center AC({Bs}s∈A,C) is a nonempty, τ-sequentially compact set. If Cis convex, so is AC({Bs}s∈A,C). We now prove the following technical lemma. Lemma. LetC beaconvexboundedsubsetofX.Let{Bs}s∈Abeadecreasingnetofsubsets of C such that AC({Bs}s∈A,C)=C.If C is (norm)separable,then there exists {xn}⊂Csuch that lim nxn–x=r for every x ∈C,where rdenotes the asymptotic radius of C with respect to the net {Bs}s. Proof WewilluseasimilarargumenttothatofTheoremin[]. Let {yn}be a dense sequence in Cand define yn=n i= yi n. Since y∈C=AC({Bs}s,C), we can find s∈Ssuch that r≤rs(y)≤r+. Then select any x∈Bssuch that x–y≥r–. Assume we have constructed x,x,...,xn– such that for all ≤j≤k≤n–wehave r– k+ k≤xk–yj≤r+ k. Take sn∈Ssuch that rsn(yi)≤r+ n,i=,...,nand rsn(yn)≤r+ n.Selectxn∈Bsn such that r– n≤xn–yn. Fix k≤n. We then have the following inequalities: r– n≤xn–yn≤ n  i= xn–yi n =xn–yk n+n  i=,i=k xn–yi n ≤xn–yk n+n  i=,i=k r+ n n =xn–yk n+n– nr+ n. From this we obtain that r n– n+ n≤xn–yk nand it follows that r– n+ n≤xn–yk≤rsn(yk)≤r+ n. Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 6 of 19 Thus for a fixed k, it easily follows limn→∞xn–yk=r.Since{yn}is dense in C,we deduce limn→∞ xn–x=rfor all x∈C. Recall that every left reversible semitopological semigroup Sbecomes a directed set when the following partial order is defined: a,b∈S,a≥b⇐⇒ aS ⊂cl(bS), where cl(bS) denotes the topological closure of the right ideal bS. Let Cbe subset of X,Sbe a left reversible semitopological semigroup, and consider a nonexpansive action of Sacting on C.Forafixedelementu∈Cdefine Ws=cl(sS(u)), where here the closure is taken for the norm topology. The sets {Ws:s∈S}form a nondecreasing family of subsets of C.Inthiscasedefiner(x)=limsr(x,Ws). Moreover, rts(tx)≤rs(x) for all t,s∈Sand x∈C.Indeed rts(tx)= sup y∈Wts tx–y=sup y∈tsS(u)tx–y=sup p∈Stx–tsp(u) ≤sup p∈Sx–sp(u)≤sup y∈Ws(x)=rs(x). Therefore r(tx)=infsrs(tx)≤infsrs(x)=r(x)foreveryt∈S, and this implies that the set C(λ):=x∈C:r(x)≤λ is either empty or S-invariant for every λ>. As a consequence, the following lemma is known. Lemma. Let X be aBanachspace endowed with atopology τ.Let C be a closed convex bounded subset of X which is τ-sequentially compact.Let S be a left reversible semitopological semigroup and consider a nonexpansive action of S on the set C.Assume that the function r(·)is τ-slsc.Then AC{Ws}s∈S,C is a nonempty closed convex τ-sequentially compact set which is S-invariant. Next we obtain fixed point results by means of the τ-GGLD and the Opial property. Theorem . Let X be a Banach space and τbe a topology on X.Let C be a (norm)separable closed,convex,bounded,τ-compactandτ-sequentially compact subset of X.Let S be a left reversible semigroup generating a nonexpansive action over C.Assume that for some u ∈Cthepreviousfunctionr(·)is τ-slsc.If X verifies either the τ-GGLD property or the Opial condition with respect to τ,then Fix(S)=∅.In case of the weak topology,the separability of C is not necessary. Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 7 of 19 Proof Let Fbe the family of nonempty, convex, τ-closed and S-invariant subsets of C. OrderingthefamilybyinclusionandusingZorn’slemma,weobtainasetwhichisminimal with respect to being nonempty, convex, τ-closed and S-invariant. We can then assume that Cis the minimal set. Since AC({Ws}s,C)isalsoanonempty,convex,τ-closed and S-invariant subset of C, we have that AC({Ws}s∈S,C)=C.Letrdenote the asymptotic radius of Cwithrespectto {Ws}s∈Sand take {xn}nas in Lemma .. By Theorem III.. of []wecanfurtherassume that limn,m,n=mxn–xmexists and it must then be equal to r.SinceCis τ-sequentially compact, we can assume that {xn}nis τ-convergent, say to some x∈C.Wehavethat limnxn–x=r, which contradicts the τ-GGLD property since τ-limn(xn–x)=.On the other hand, for some y∈Cwith y=x,weobtain r=lim nxn–x=lim nxn–x–(y–x)=r, which contradicts the Opial condition. Therefore Fix(S)=∅. In case that τcoincides with the weak topology, it is known that both the w-GGLD condition and the Opial condition for the weak topology imply weak normal structure [] and therefore the w-FPP for left reversible semigroups [].  Notice that for separable Banach spaces and topologies τweaker than the norm topology,theτ-compactness ofthedomain is asuperfluousassumption.Indeed,theseparability of Ximplies that Xis Lindelöf for the norm and so does for the topology τsince it is weakerthatthenormtopology.Thus, τ-sequentially compactsets arecountably compact and Lindelöf, so they are τ-compact. ManyexamplesofBanachspacesareknowntosatisfytheGGLDconditionortheOpial property with respect to some classical topologies. However, to apply Theorem .,the τ-sequentiallowersemicontinuityofthefunctionr(·)forsomeu∈Cmustbe checked.In what follows we study equivalent and sufficient conditions to assure this statement. Let{xn}beaboundedsequence.Wedefinethetypefunctionassociatedtothesequence {xn}nby (x)=limsup nx–xn,x∈X. In case that τ-limnxn=wesaythatis a τ-null type function. Recall that given {Bs}s∈Aa decreasing net of bounded subsets of Xwe defined r(·)associated to the net {Bs}s∈Aas r(x)=limsr(x,Bs). The following lemma will be very helpful to assure whether the function r(·)isτ-sequentially lower semicontinuous. Lemma. The function r(·)is τ-slsc if and only if the type functions (·)are τ-slsc. Proof One implication is direct since we can take Bs={xn:n≥s}.Then(x)=r(x)= limsupnxn–x. Assume that the type functions are τ-sequentially lower semicontinuous. Take {Bs}s∈A any decreasing net of bounded subsets, and let (yn)nbe a τ-convergent sequence to some point y∈X.Wehavetoprovethatr(y)≤liminfnr(yn). Consider a sequence (n)nof positive real numbers with limnn=. Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 8 of 19 We claim that there exists a sequence {xn}nwith xn∈Bsnand sn>···>s,suchthat r(y)–n≤y–xn<r(y)+n,() yi–xn≤r(yi)+n;i=,,...n.() Indeed,take s∈Asuch that r(y,Bs)<r(y)+;r(y,Bs)<r(y)+ and x∈Bswith r(y)–<y–x<r(y)+. Assume now that we have obtained x,...,xnsatisfying ()and(). Take sn+ >snsuch that r(y,Bsn+)<r(y)+n+ and r(yi,Bsn+)<r(yi)+n+ for i=,...,n,n+. Consider xn+ ∈Bsn+ with r(y)–n+ <y–xn+<r(y)+n+. Moreover, yi–xn+≤r(yi,Bsn+)≤r(yi)+n+;i=,,...,n+, and the claim is proved. By (), limsupnxn–y=r(y)andby(), limsupnxn–yi≤r(yi)foreveryi∈N. We consider the type function (x)=limsupnxn–x.Wenowhave r(y)=limsup nxn–y=(y)≤liminf i(yi) =liminf ilimsup nxn–yi≤liminf ir(yi), which implies that the function r(·)isτ-sequentially lower semicontinuous as we wanted to prove.  InthesettingofTheorem.,thesetCisτ-sequentiallycompact,sowecanassumethat the sequence {xn}obtained in the previous lemma is τ-convergent. Moreover, since the topology is translation invariant, we only need to assume that the τ-null type functions are τ-sequentially lower semicontinuous. Finally, we can state our main result in this section. Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 9 of 19 Theorem. LetX beaseparableBanachspace,τbeatranslationinvarianttopologyon Xsuchthatτ-compact sets are τ-sequentially compact.Assume that the τ-null functions are τ-slsc.If X verifies either the τ-GGLD property or the τ-Opial condition,Xhasthe τ-FPP for left reversible semigroups. Notice that when τis the weak topology, the separability of Xis not necessary. On the other hand, it is said that the τ-null type functions are constant on spheres if limsup nxn–x=limsup nxn–y for every x,y∈Xwith x=y,where{xn}nis a norm bounded τ-null sequence. In [] (Lemma ) it is proved that the norm and the τ-null type functions are τ-slsc whenever they are constant on spheres. 4 First examples and applications for the weak-star topology InthissectionwearegoingtoapplyTheorem.toseveraldifferentclassesofdualBanach spaces endowed with their weak∗topologies. To begin with, consider X=(,·), where by ·we denote the usual norm, and let τbetheweak∗topology σ(,c), which is metrizable. It can easily be checked that for every w∗-null sequence {xn}nand for all x∈, limsup nxn+x=x+limsup nxn, which implies both the w∗-GGLD property and the w∗-sequential lower semicontinuity of the w∗-null type functions. Therefore we deduce that has the weak∗-FPP for left reversiblesemigroups[].Thisresultcanbegeneralizedasfollowssincethesameequality holds for w∗-null sequences. Corollary. Let (Xn)nbe a sequence of finite dimensional Banach spaces.Then the onedirect sum X=⊕ n∈NXn has the weak∗-FPP for left reversible semigroups,where the considered predual is defined by E ={x=(xn)n:xn∈Xn,limnxnXn=}. IncasethatGisaseparablecompactgroup,itsFourier-StieltjesalgebraB(G)isthedirect one-sum of a sequence of finite dimensional Banach spaces (see Section  in []and Chapter I, Theorem . in []). Applying Corollary . we can deduce the following. Corollary . [] Let G be a separable compact group and B(G)be its Fourier-Stieltjes algebra.Then B(G)satisfies the w∗-FPP for left reversible semigroups. Wecannowstateanimprovementoftheresultsin[]aboutthew∗-FPP for left reversible semigroups as follows. Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 16 of 19 We finish the section with the Banach space introduced in Example .,forwhichwe showedthattheasymptoticcenterswerenotweaklycompactingeneral,sothearguments used in [, , ]or[] are not valid for proving the w∗-FPP for left reversible semigroups. Example. Let X:=(,·), where the norm ·is defined as x=max n∈Nx(n)+  ∞  i=n+x(i). Here  x≤x≤ xfor every x∈.Noticethatifx,yare vectors in such that max{supp(x)}<min{supp(y)}, then it is not difficult to check that x+y=maxy,x+ y.() The previous equality also proves that the Schauder basis is monotonous for ·and Xis isometric to a dual space. Assume that {xn}is a weak∗-null sequence, where by w∗topology we mean the σ(,c) topology. Without loss of generality, we can assume that {xn}is a block basic sequence, l=limnxnexists and that limn,m;n=mxn–xm= limsupnlimsupmxn–xm. Therefore limsup nlimsup mxn–xm=limsup nlimsup mmaxxm,xn+ xm =limsup nmaxl,xn+ limsup mxm =l+ limsup mxm≥ l, which implies the w∗-GGLD property. Let us prove that the weak∗-null type functions are w∗-lower semicontinuous. Take {xn}aw∗-null sequence and y=∞ n= ynen∈.Wecanassumethat{xn}is ablock basic sequence. Let >ands∈Nsuch that y–ys<for every s≥s,whereys= s n= ynen.Noticethat(y)=limsupny–xn=limslimsupnys–xn. Moreover, from the definition of the norm and from the equality (), limsup nys–xn=maxlimsup nxn,ys+ limsup nxn ≤maxlimsup nxn,y+ limsup nxn for every s∈N. On the other hand, for every s≥s, maxlimsup nxn,y+ limsup nxn ≤maxlimsup nxn,ys+ limsup nxn+ =limsup nys–xn+. Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 17 of 19 Taking limits when sgoes to infinity, we deduce that for every y∈, limsup ny–xn=maxlimsup nxn,y+ limsup nxn. The above implies that the w∗-null type functions are constant on spheres and therefore w∗-slsc. Therefore (,·)verifiesthew∗-FPP for left reversible semigroups Notice also that this space fails the Opial condition with respect to the w∗topology.Indeed,consider the weak∗-null sequence xn=enfor n∈Nand the vector x= e.Thenlimnxn= limnxn+x. 5 Modular function spaces and the ρ-FPP for left reversible semigroups In this section we consider modular function spaces Lρ:= {f∈M:ρ(αf)→asα→} endowed with the Luxemburg norm f:=infα>:ρf α≤, and the Orlicz norm fo:=inf k+ρ(kf):k>. Here Mdenotes a set of measurable functions and ρa convex additive function modular defined over M.Itissaidthatasequence(fn)⊂Lρconverges to fρ-almost everywhere, fn→fρ-a.e., if {w∈:f(w)=limnfn(w)}is ρ-null. For all general definitions we refer to [,]or[]. A function modular ρis said to satisfy the -type condition if there exists some K> such that ρ(f)≤Kρ(f)foreveryf∈Lρ. In this section we assume that ρis a σ-finite convex additive function modular which satisfies the -type condition (see []or[]). In this case, a topology τρis defined overLρsuchthatτρ-compactsetscoincideexactlywithτρ-sequentiallycompactsets[]. Moreover, the ρ-convergence coincides with the τ-convergence up to subsequences. In[] (Section ) it is proved that, under the -type condition, the modular function space Lρverifies the τρ-uniform Opial condition for both the Luxemburg and the Orlicz norm. Moreover, Lemma . and Lemma . of [] show that the τρ-null type functions are τρ-sequentially lower semicontinuous. Hence we can state the following theorem. Theorem . Let ρbe a σ-finite convex additive function modular which satisfies the -condition.Then the modular function space Lρverifies the τρ-FPP for left reversible semigroups when it is equipped with either the Luxemburg or the Orlicz norm. Examples of modular function spaces are the Lp-spaces and more generally the Musielak-Orlicz function spaces, where the τρtopology coincides with the local convergence in measure topology. For the definition of Musielak-Orlicz function spaces and for more examples of modular function spaces, see []and[]. Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 18 of 19 6L-Embedded Banach spaces and the FPP for left reversible semigroups with respect to the abstract measure topology In this section we consider L-embedded Banach spaces endowed with the so-called measure topology. A Banach space Xissaid tobe an L-embeddedBanach spaceif thereexists aclosedsubspaceXs⊂X∗∗ such that X∗∗ =X⊕Xs. A wide study and many examples of thisclassofBanachspacescanbefoundinthemonograph[].ExamplesofL-embedded Banach spaces are the following: () Duals of M-embedded Banach spaces. () L(μ)-spaces and preduals of von Neumann algebras. Recallthata sequence {xn}is said tospan anasymptotically isometriccopy of ifthere exists a nonincreasing sequence {δn}⊂[,) tending to  such that ∞  n= (–δn)|αn|≤ ∞  n= αnxn≤∞  n= |αn| for every {αn}∈.Inthiscasewewilldenotexn∼(asy). For L-embedded Banach spaces, the abstract measure topology (τμ)isdefinedin[] (Section)byconsideringtheclassofconvergentsequences.Namely,if{xn}isasequence inanL-embeddedBanachspace,wesaythat{xn}tendstointheabstractmeasuretopology (τμ-limnxn=)if {xn}is norm bounded and every subsequence {xnk}contains a subsequence {xnkl} such that xnkl/xnkl∼(asy)or xnkl→. When Xis a separable L-embedded Banach space, the notions of compactness and sequential compactness agree for τμ[]. It is proved in [](seealso[])thatforeveryτμ-nullsequence{xn}inan L-embedded Banachspace, limsup nxn+x=limsup nxn+x for all x∈X. This equality implies the τμ-GGLD property and the τμ-sequential lower semicontinuity of the τμ-null type functions. It is known that L-embeddedBanachspacessatisfytheFPP fornonexpansivemappings with respect to the abstract measure topology []. According to Theorem .,wecan extend this result to left reversible semigroups in the following way. Corollary. LetX beaseparableL-embeddedBanachspace.ThenX verifiestheτμ-FPP for left reversible semigroups. As a particular case we can deduce Theorem . in [] when the Hilbert space His separable. Indeed, in case that the L-embedded Banach space is L(M,τ) for some finite von Neumann algebra Mdefined over a Hilbert space, the previous measure topology coincides with the usual measure topology defined on L(M,τ)forboundedsets(seeTheorem . in []). Hence, noncommutative L-spaces verify the fixed point property for left reversible semigroups with respect to the usual measure topology. This topology is in fact the convergence locally in measure topology in case that L(M,τ)=L(μ)forsome σ-finite measure space. Castillo-Santos and Japón Fixed Point Theory and Applications (2015) 2015:109 Page 19 of 19 Competing interests The authors declare that they have no competing interests. Authors’ contributions Both authors have contributed equally and significantly in writing this article. Both authors read and approved the final manuscript. Author details 1CIMAT, Centro de Investigación en Matemáticas, CONACYT, Consejo Nacional de Ciencias y Tecnología, Guanajuato, México. 2Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Tarfia s/n, Sevilla, 41012, Spain. Acknowledgements The authors would like to thank the referees for their valuable suggestions to improve the presentation of this article. The second author is partially supported by MCIN, Grant MTM-2012-34847-C02-01 and Junta de Andalucía, Grants FQM-127 and P08-FQM-03543. Received: 27 January 2015 Accepted: 18 June 2015 References 1. Paterson, ALT: Amenability. Mathematical Surveys and Monographs, vol. 19. Am. 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