R E S E A R C H
Open Access
The
τ
-fixed point property for left
reversible semigroups
Francisco E Castillo-Santos
1and Maria A Japón
2**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
A semigroupSis said to be a semitopological semigroup ifSis equipped with a Hausdorff topology such that for eacha∈S, the two mappings fromSintoSdefined bys→asand s→saare continuous. A semitopological semigroupSis said to be left reversible if any two nonempty closed right ideals ofShave 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 [].
LetCbe a subset of a Banach spaceXand letSbe a semitopological semigroup. A non-expansive action ofSon the setCis a mapφ:S×C→C, denoted byφ(s,u) =s(u) (orsu), which satisfies:
(i) ts(u) =t(su)for allt,s∈Sandu∈C.
(ii) For allu∈C, the functions∈S→s(u)∈Cis continuous.
(iii) For everys∈S, the mappingu∈C→s(u)∈Cis nonexpansive.
A subset C is said to verify the fixed point property for left reversible semigroups if for every left reversible semitopological semigroupSand for every nonexpansive action
φ:S×C→C, the setFix(S) :={u∈C:t(u) =u,∀t∈S}is nonempty.
Definition . LetXbe a Banach space andτ be a topology onX. It is said thatXhas theτ fixed point property (τ-FPP) for left reversible semigroups if every closed, convex, bounded subsetCwhich isτ-compact has the fixed point property for left reversible semi-groups.
Given a nonexpansive mappingT, if we replace the left reversible semigroup by the dis-crete and Abelian semigroup{T,T,T, . . .}acting fromCtoC, Definition . becomes the
usual definition of theτ-FPP for nonexpansive mappings. There exist some Banach spaces failing thew-FPP [] and therefore they fail thew-FPP for left reversible semigroups (we can consider the semigroupS={T,T,T, . . .}whereTis the fixed point free
nonexpan-sive 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 set-ting of nonexpansive actions of left reversible semigroups, that is, weak normal structure implies thew-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 algebraB(G) of a separable compact group ver-ifies the weak∗-FPP for left reversible semigroups. Notice that ifGis the torus group, then B(G) is isometric to(Z). In , Randrianantoanina [] proved that the spaceT(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 gen-eral dual Banach spaces since they are mainly based on the following fact: in the above-mentioned 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 com-pact. 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 spaceL[, ] or, more generally,
every noncommutativeL-space associated to a finite von Neumann algebra satisfies the
fixed point property for left reversible semigroups with respect to the abstract measure topologyτ(the convergence in measure topology in case ofL[, ]). Here the asymptotic
centers ofτ-compact sets are norm compact.
In this paper we develop new arguments to deduce whether a dual Banach space satis-fies the weak∗-FPP for left reversible semigroups. More generally, we will considerτas any translation invariant topology on a separable Banach spaceXand we give sufficient con-ditions to assure theτ-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 theτ-FPP for left reversible semigroups. Here we will consider dif-ferent 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
non-isomorphic to. We will also consider the topology of the convergence locally in measure
in some function spaces, the abstract measure topology inL-embedded Banach spaces and the topology ofρ-almost everywhere convergence in modular function spaces.
2 Preliminaries
We introduce some definitions and concepts.
LetXbe a Banach space and let{Bs}s∈Abe a decreasing net of bounded subsets ofX. For
x∈Xands∈A, we consider
rs(x) :=sup
x–y:y∈Bs
,
r(x) :=infrs(x) :s∈A
=lim
s rs(x).
Notice thatr(·) is a continuous function for the norm topology.
We defined the asymptotic radius and the asymptotic center of a setCwith 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 . LetXbe the spacerenormed as follows:
x=max
n∈N
x(n)+
∞
i=n+
x(i) .
Since{en}is a monotonous boundedly complete Schauder basis forX= (,·), this space
is isometric to the dual of the closed subspace spanned by the orthogonal functions{e∗n}. ThusXis a dual space and in fact its weak∗topology coincides with theσ(,c) topology.
Consider the set
C:= ∞
n=
tnen:tn≥,
∞
n=
tn≤
,
which is a closed convex boundedw∗-compact set.
Define the bounded subsetsBs={ek:k≥s}, where the{en}are the unit basic vectors. In
this caser(x) =lim supsx–es, and it is easy to check thatr(x)≥ for allx∈. However,
for allk,nwithk<n,
ek–en
= ,
which implies that enbelongs to the asymptotic center ofCwith respect to the sequence
{Bs}s, and this center is not weakly compact. Therefore, the arguments used in [–] or []
to prove the existence of a common fixed point for left reversible semigroups of nonex-pansive mappings are not useful in this example. We will later deduce that (, · ) does
satisfy thew∗-FPP for left reversible semigroups.
withτ-limntn=t. From now on, letXbe a Banach space and letτbe a translation
invari-ant topology onX, that is,τ-limnxn=xif and only ifτ-limn(xn–x) = .
We introduce the following definitions which generalize two geometric properties which 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 () [] whenτ is the weak topol-ogy.
Definition . Let (X,·) be a Banach space andτbe a topology onX. We will say thatX has the generalized Gossez-Lami Dozo property for the topologyτ(τ-GGLD) if for every (norm) bounded andτ-null sequence{xn}such thatlimxn = andlimn,m,n=mxn–xm
exists, it is the case that
limxn< lim
n,m,n=mxn–xm.
Definition . It is said that a Banach space (X, · ) satisfies the Opial condition with respect to a topologyτif
lim inf
n xn–x<lim infn xn–x
for allx∈Xwithx=x, whenever (xn) is a sequence inXwithτ-limnxn=x.
Theτ-GGLD property and the Opial condition will be the key to our main results. It is well known that these properties are not related. We will also illustrate this assertion with some examples.
Definition . Let (X, · ) be a Banach space andτbe a topology onX. It is said thatXis uniformly Kadec-Klee with respect toτ,UKK(τ), if for every> there exists someδ> such that whenever{xn}nis a sequence in the closed unit ball ofX, which isτ-convergent
to a pointx∈Xwithinfn=mxn–xm>, thenx< –δ.
Associated with theUKK(τ) property, the following modulus is defined:
PX,τ() =inf
–x:xn ≤,τ-lim
n xn=x,ninf= mxn–xm>
.
In case that τ is the weak topology the previous coefficient is known as Partington’s modulus, and it is clear that a Banach space has theUKK(τ) property if and only ifPX,τ() >
for every∈(, ]. We denotePX,τ(–) =lim→–PX,τ(). Forτa linear topology, it is not
difficult to check that
limxn ≤
–PX,τ
– lim
n,m,n=mxn–xm
for every (norm) bounded τ-null sequence {xn} such that limn,m,n=mxn–xm exists.
3 Main results
LetCbe a set and{Bs}s∈Abe a decreasing net of bounded subsets ofX. It is clear that
AC{Bs}s∈A,C
=
n∈N
x∈C:r(x)≤r+
n
.
Therefore, if the set C is τ-sequentially compact and the functionr(·) is τ-slsc, the asymptotic centerAC({Bs}s∈A,C) is a nonempty,τ-sequentially compact set. IfCis
con-vex, so isAC({Bs}s∈A,C).
We now prove the following technical lemma.
Lemma . Let C be a convex bounded subset of X.Let{Bs}s∈Abe a decreasing net of subsets
of C such thatAC({Bs}s∈A,C) =C.If C is(norm)separable,then there exists{xn} ⊂C such
that
lim
n xn–x=r
for every x∈C,where rdenotes the asymptotic radius of C with respect to the net{Bs}s.
Proof We will use a similar argument to that of Theorem in []. Let{yn}be a dense
sequence inCand defineyn=ni=yi
n.
Sincey∈C=AC({Bs}s,C), we can finds∈Ssuch that
r≤rs(y)≤r+ .
Then select anyx∈Bssuch thatx–y ≥r– .
Assume we have constructedx,x, . . . ,xn–such that for all ≤j≤k≤n– we have
r–
k+
k≤ xk–yj ≤r+
k.
Takesn∈Ssuch thatrsn(yi)≤r+
n,i= , . . . ,nandrsn(yn)≤r+n. Selectxn∈Bsn
such thatr–n ≤ xn–yn.
Fixk≤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 thatr
n –
n+n ≤
xn–yk
n and it follows that
r–
n+
n≤ xn–yk ≤rsn(yk)≤r+
Thus for a fixedk, it easily followslimn→∞xn–yk=r. Since{yn}is dense inC, we
deducelimn→∞xn–x=rfor allx∈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),
wherecl(bS) denotes the topological closure of the right idealbS.
LetCbe subset ofX,Sbe a left reversible semitopological semigroup, and consider a nonexpansive action ofS acting onC. For a fixed element u∈Cdefine Ws=cl(sS(u)),
where here the closure is taken for the norm topology. The sets{Ws:s∈S}form a
non-decreasing family of subsets ofC. In this case definer(x) =limsr(x,Ws). Moreover,
rts(tx)≤rs(x)
for allt,s∈Sandx∈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).
Thereforer(tx) =infsrs(tx)≤infsrs(x) =r(x) for everyt∈S, and this implies that the set
C(λ) :=x∈C:r(x)≤λ
is either empty orS-invariant for everyλ> . As a consequence, the following lemma is known.
Lemma . Let X be a Banach space endowed with a topologyτ.Let C be a closed convex bounded subset of X which isτ-sequentially compact.Let S be a left reversible semitopo-logical semigroup and consider a nonexpansive action of S on the set C.Assume that the function r(·)isτ-slsc.Then
AC{Ws}s∈S,C
is a nonempty closed convexτ-sequentially compact set which is S-invariant.
Next we obtain fixed point results by means of theτ-GGLD and the Opial property.
Proof LetF be the family of nonempty, convex,τ-closed andS-invariant subsets ofC. Ordering the family by inclusion and using Zorn’s lemma, we obtain a set which is minimal with respect to being nonempty, convex,τ-closed andS-invariant. We can then assume thatCis the minimal set.
SinceAC({Ws}s,C) is also a nonempty, convex,τ-closed andS-invariant subset ofC,
we have thatAC({Ws}s∈S,C) =C. Letrdenote the asymptotic radius ofCwith respect to
{Ws}s∈Sand take{xn}nas in Lemma .. By Theorem III.. of [] we can further assume
thatlimn,m,n=mxn–xmexists and it must then be equal tor. SinceCisτ-sequentially
compact, we can assume that{xn}n isτ-convergent, say to somex∈C. We have that
limnxn–x=r, which contradicts theτ-GGLD property sinceτ-limn(xn–x) = . On
the other hand, for somey∈Cwithy=x, we obtain
r=lim
n xn–x=limn xn–x– (y–x)=r,
which contradicts the Opial condition. ThereforeFix(S)=∅.
In case that τ coincides with the weak topology, it is known that both thew-GGLD condition and the Opial condition for the weak topology imply weak normal structure [] and therefore thew-FPP for left reversible semigroups []. Notice that for separable Banach spaces and topologiesτ weaker than the norm topol-ogy, theτ-compactness of the domain is a superfluous assumption. Indeed, the separabil-ity ofXimplies thatXis Lindelöf for the norm and so does for the topologyτ since it is weaker that the norm topology. Thus,τ-sequentially compact sets are countably compact and Lindelöf, so they areτ-compact.
Many examples of Banach spaces are known to satisfy the GGLD condition or the Opial property with respect to some classical topologies. However, to apply Theorem ., the
τ-sequential lower semicontinuity of the functionr(·) for someu∈Cmust be checked. In what follows we study equivalent and sufficient conditions to assure this statement.
Let{xn}be a bounded sequence. We define the type function associated to the sequence
{xn}nby
(x) =lim sup
n x
–xn, x∈X.
In case thatτ-limnxn= we say thatis aτ-null type function.
Recall that given{Bs}s∈Aa decreasing net of bounded subsets ofXwe definedr(·)
asso-ciated to the net{Bs}s∈Aasr(x) =limsr(x,Bs). The following lemma will be very helpful to
assure whether the functionr(·) isτ-sequentially lower semicontinuous.
Lemma . The function r(·)isτ-slsc if and only if the type functions(·)areτ-slsc.
Proof One implication is direct since we can takeBs={xn:n≥s}. Then(x) =r(x) =
lim supnxn–x.
Assume that the type functions areτ-sequentially lower semicontinuous. Take{Bs}s∈A
any decreasing net of bounded subsets, and let (yn)nbe aτ-convergent sequence to some
pointy∈X. We have to prove thatr(y)≤lim infnr(yn).
We claim that there exists a sequence{xn}nwithxn∈Bsnandsn>· · ·>s, such that
r(y) –n≤ y–xn<r(y) +n, ()
yi–xn ≤r(yi) +n; i= , , . . .n. ()
Indeed, takes∈Asuch that
r(y,Bs) <r(y) +; r(y,Bs) <r(y) +
andx∈Bswith
r(y) –<y–x<r(y) +.
Assume now that we have obtainedx, . . . ,xnsatisfying () and (). Takesn+>snsuch
that
r(y,Bsn+) <r(y) +n+
and
r(yi,Bsn+) <r(yi) +n+
fori= , . . . ,n,n+ . Considerxn+∈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 (),lim supnxn–y=r(y) and by (),lim supnxn–yi ≤r(yi) for everyi∈N.
We consider the type function(x) =lim supnxn–x. We now have
r(y) =lim sup
n
xn–y=(y)≤lim inf i (yi)
=lim inf
i lim supn xn–yi ≤lim infi r(yi),
which implies that the functionr(·) isτ-sequentially lower semicontinuous as we wanted
to prove.
In the setting of Theorem ., the setCisτ-sequentially compact, so we can assume that the sequence{xn}obtained in the previous lemma isτ-convergent. Moreover, since the
topology is translation invariant, we only need to assume that theτ-null type functions areτ-sequentially lower semicontinuous.
Theorem . Let X be a separable Banach space,τ be a translation invariant topology on X such thatτ-compact sets areτ-sequentially compact.Assume that theτ-null functions are τ-slsc.If X verifies either theτ-GGLD property or theτ-Opial condition,X has the
τ-FPP for left reversible semigroups.
Notice that whenτ is the weak topology, the separability ofXis not necessary. On the other hand, it is said that theτ-null type functions are constant on spheres if
lim sup
n xn
–x=lim sup
n xn
–y
for everyx,y∈Xwithx=y, where{xn}nis a norm boundedτ-null sequence. In []
(Lemma ) it is proved that the norm and theτ-null type functions areτ-slsc whenever they are constant on spheres.
4 First examples and applications for the weak-star topology
In this section we are going to apply Theorem . to several different classes of dual Banach spaces endowed with their weak∗topologies.
To begin with, considerX= (, · ), where by · we denote the usual norm, and
letτ be the weak∗topologyσ(,c), which is metrizable. It can easily be checked that for
everyw∗-null sequence{xn}nand for allx∈,
lim sup
n xn
+x=x+lim sup
n xn
,
which implies both thew∗-GGLD property and thew∗-sequential lower semicontinuity of thew∗-null type functions. Therefore we deduce that has the weak∗-FPP for left
reversible semigroups []. This result can be generalized as follows since the same equality holds forw∗-null sequences.
Corollary . Let(Xn)nbe a sequence of finite dimensional Banach spaces.Then the
one-direct sum
X=⊕
n∈N Xn
has the weak∗-FPP for left reversible semigroups,where the considered predual is defined by E={x= (xn)n:xn∈Xn,limnxnXn= }.
In case thatGis a separable compact group, its Fourier-Stieltjes algebraB(G) is the direct 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.
Theorem . Let(X, · )be a Banach space with a boundedly complete Schauder basis {en}n.Assume that for every block basic sequence{xn}nand for every x∈X,there holds
lim sup
n
x+xn=x+lim sup n
xn
for every x∈X.Then X has the w∗-FPP for left reversible semigroups where w∗=σ(X,Z) and Z is the closed subspace spanned by the orthogonal functions{e∗n}n.
Proof When the Schauder basis is boundedly complete, the Banach spaceXis isomorphic toZ∗([], Proposition .b..), and we can consider the weak∗topologyσ(X,Z) where the convergence is equivalent to the pointwise convergence for bounded sequences. Now the w∗-null type functions arew∗-sequentially lower semicontinuous since they are constant on spheres and thew∗-GGLD property is satisfied. We can apply Theorem . to more general Banach spaces. The following example is given in [].
Example . Letcbe the vector space of all sequences of scalars which are eventually
zero. Denote by [N]<w the subsets ofNwith finite cardinality. ForA⊂N, letP
Abe the
usual projection, that is, ifx= (x(n))n∈c,PA(x) is the vector whose coordinates arex(n)
ifn∈Aand zero otherwise. Consider the familyS⊂[N]<wby
S:=S= (n, . . . ,nk)∈[N]<w:ni+≥nifori= , . . . ,k–
.
Define
|x|:=sup
S∈S
PS(x),
and letXbe the completion ofcwith the norm|·|. Then (X,|·|) is a dual Banach space
satisfying the hypotheses of Theorem .. Indeed, the sequence{en}forms a monotonous
bounded complete Schauder basis, which implies that Xis isometric to the dual of Z, whereZ is the Banach space spanned by the orthogonal functionals{e∗n}. The fact that Xsatisfies the equality given in Theorem . is proved in []. Therefore, it satisfies the w∗-FPP for left reversible semigroups. Notice that this Banach space is not isomorphic to
[].
Corollary .[] Let H be a Hilbert separable Banach space.ThenT(H),the space of the trace class operators on H,has the weak∗-FPP for left reversible groups,where the predual is E=K(H),the space of all compact operators defined on H.
In this case, Lennard proved thatT(H) verifies the weak∗ uniform Kadec-Klee prop-erty [] and therefore thew∗-GGLD condition. Proposition . in [] implies that the w∗-null type functions arew∗-sequentially lower semicontinuous. (Notice that in [] the separability of the Hilbert space can be dropped.)
Notice thatH() is a separable Banach space, it has thew∗-GGLD property since it
verifies thew∗-UKK condition [], and Lemma . in [] implies that thew∗-null type functions arew∗-sequentially lower semicontinuous. Here, the weak∗ topology refers to the isometric predualC(T)/A(T), whereC(T) is the space of all continuous functions on
the torusT, with the usual supremum norm, andA(T) is the set of boundary values of
the disc algebra with zero constant term.
Recall that mapφ: [, +∞)→[, +∞) is said to be an Orlicz function ifφis convex, vanishing at , continuous and not identically equal to zero. A sequence :={φn}n of
Orlicz functions is called a Musielak-Orlicz function.
Given a Musielak-Orlicz function, a convex modularIis defined on the set of all real
sequences given by
I(x) =
∞
n=
φnx(n).
A Musielak-Orlicz sequence space generated byis defined by
=x= (xn) :I(λx) < +∞for someλ>
.
We can considerequipped with the Luxemburg norm
x:=infk> :I(x/k)≤
,
or with the Orlicz norm
xo:=inf
k
+I(kx)
:k>
.
It is well known that both norms are equivalent andis a Banach space [, ]. In case thatφm=φnfor alln,m∈N, we simply say thatis an Orlicz sequence Banach
space.
It is said that a Musielak-Orlicz function={φn}nsatisfies the conditionδif there are
positive constantsaandKand a nonnegative sequence (cn)∈such that
φn(t)≤Kφn(t) +cn
for everyn∈N,t∈R+satisfyingφn(t)≤a.
For Orlicz sequence Banach spaces, it is said that an Orlicz functionφsatisfies the con-ditionδif there exist somet> andK> such thatφ(t)≤Kφ(t) for everyt∈[,t].
When the conditionδis satisfied, the unit vectors form a boundedly complete
normal-ized unconditional basis of ([], Proposition .d.). We denote bye∗nthe functional vector associated withenfor everyn∈Nand consider [e∗n] the closed span of the vectors
{e∗n}in the dual of the Musielak-Orlicz space. Since the Schauder basis is monotone, the dual of the Banach space [e∗n] is isometric toφ([], Proposition .b..). Therefore we can
consider the weak∗topologyσ(, [e∗n]). At this point we can state the following.
Theorem . Let ={φn} be a Musielak-Orlicz function satisfying the condition δ.
Proof Notice thatis a separable Banach space. The arguments included in the proof of (iii)⇒(i) of Theorem in [] imply thatendowed with the Luxemburg norm satis-fies thew∗-uniform Opial condition. From the proof of Theorem in [] we can deduce that the Musielak-Orlicz spacesatisfies thew∗-uniform Kadec-Klee property when it
is equipped with the Orlicz norm and so thew∗-GGLD property. Since the weak∗ conver-gence implies the coordinatewise converconver-gence, standard arguments show that
lim sup
n
I(xn+x) =lim sup n
I(xn) +I(x)
for everyx∈and for everyw∗-null sequence{xn}. This implies that the functionalx∈
→lim supnIφ(xn–x) isw∗-slsc. With this property, it is easy to check thew∗-sequential
lower semicontinuity for thew∗-null type functions for both norms, the Luxemburg and
the Orlicz norm.
Particular examples of Musielak-Orlicz sequence spaces are the Nakano spaces (pn)
where the Orlicz functions are φn(t) = |t|pn with pn⊂ [, +∞) []. In this case, the
Musielak-Orlicz function satisfies the conditionδ if and only ifsupnpn< +∞. In case
thatpn= for everyn∈N, we obtain. However, there exist some dual Nakano spaces
which are not isomorphic toand satisfyinglimnpn= [].
Corollary . Let(pn)be a bounded sequence in[, +∞).Then the corresponding Nakano
sequence Banach space verifies the w∗-FPP for left reversible semigroups.
In the rest of this section we will consider some relevant equivalent norms in, and
we will study whether they provide the w∗-FPP for left reversible semigroups. On the one hand, there are some equivalent norms in which fail to have this property. For
instance endowed with the normx=max{x+,x–}, as the dual ofcwith the
normx+∞+x–∞, fails thew∗-FPP for nonexpansive mappings []. On the other hand,
theτ-GGLD and the Opial conditions are properties that can be transferred by isomor-phisms under certain conditions. For instance, it is known (see []) that if we consider Y = (,| · |) for some equivalent norm, then Y satisfies theσ(,c)-GGLD condition
wheneverd(Y, (, · )) < (and this is the best upper bound that can be obtained due to
the previous norm). However, as far as we know, thew∗-FPP for left reversible semigroups cannot be derived from stability results since the same does not hold for theτ-sequential lower semicontinuity of theτ-null type functions. Indeed, consider the Euclidean norm · inRand define the equivalent norm inby
v(x) :=
x()u+
∞
n=
x(n)u
,
whereu= (, ) andu= (–a, ( –a)/) for somea∈(, ). Finally, forλ> , define the
norm
|x|λ:=x+λv(x)
for allx∈. It is not difficult to check that| · |λis equivalent to · and that the
xn=e+aen, which tends tox=ein theσ(,c)-topology. Notice that
|x|λ= +λ> +λ
–a=|x
n|λ
for everyn∈N. Hence the norm| · |λis notσ(,c)-sequentially lower semicontinuous,
and consequentlyσ(,c)-null type functions are notσ(,c)-slsc in general.
A relevant renorming in metric fixed point theory was given by Lin in [] in the follow-ing way: Let{γk}be a nondecreasing sequence of positive numbers converging to and
define
{an}:=sup
k
γk
∞
n=k
|an|
for every {an} ∈. Lin proved that endowed with this norm verifies the fixed point
property for nonexpansive mappings, that is, every| · |-nonexpansive mapping defined on a closed convex bounded subset ofinto itself has a fixed point. This assertion proves
that FPP does not imply reflexivity as it was conjectured for a long time.
At this point, we do not know whether (,| · |) verifies a similar statement for left
re-versible semigroups of nonexpansive mappings. Here we prove that (,| · |) does verify
thew∗-FPP for left reversible semigroups where by the weak∗topology we refer toσ(,c).
Notice that the above norm is a dual norm, that is, ifX= (,| · |) thenXis isometric to
a dual space. This can be deduced from the fact that the Schauder basis{en}is
bound-edly complete and it is monotonous for the| · |norm. Moreover,Xis isometric to the dual of the Banach space spanned by [e∗n] so the weak∗topology is in factσ(,c) (see
Proposition .b. in []).
If{xn}nis aw∗-null sequence in, thenlim supn|xn–xm|= lim supn|xn|[], which
clearly implies thew∗-GGLD. Notice that thew∗-null type functions are not constant in spheres: consider the sequencexn=enfor everyn∈Nandx=γe,y=γe. Now|x|=
|y|= but
lim sup
n |xn
–x|= +γ and lim sup
n |xn
–x|= +γ,
so we cannot derive thew∗-sequential lower semicontinuity of thew∗-null type functions from this fact. We will consider the following result which is due to Lennard and it was included in []. However, we here develop a shorter proof for the sake of completeness.
Lemma . Let{xn}nbe a w∗-null sequence in.Define
(y) :=lim sup
n→∞ |xn–y|.
Then
(y) =sup
k∈N
γk
∞
n=k
|yn|+lim sup n |xn|
()
for every y∈with y=
∞
Proof We may, without loss of generality, assume that the sequence{xn}is disjointly
sup-ported.
First of all, assume that the vectoryis finitely supported, that is, there exists somen∈N
such thatyn= ifn>n. We can assume thatn<min{supp(x)}. In such case, for every
n∈N,
|y–xn|=max
sup
≤k≤n
γk
n
n=k
|yn|+xn ,|xn|
.
Taking limits whenngoes to infinity and using thatlim supnxn=lim supn|xn|since
{xn}is disjointly supported, we deduce
lim sup
n |y
–xn|=max
sup
≤k≤n
γk
n
n=k
|yn|+lim sup n |xn|
,lim sup
n |xn|
=sup k∈N γk ∞
n=k
|yn|+lim sup n |xn|
,
where the last equality follows from the fact thatlimkγk= andyn= ifn>n. Then the
lemma holds in this case.
Lety=∞n=ynenbe any vector inand denoteys=
s
n=ynen. Notice thatis a
con-tinuous function for the norm topology so(y) =lims(ys). If we prove thatlims(ys)
coincides with the right part of the equality stated in the lemma, the proof will be fin-ished.
For everys∈N,
(ys) =max
sup
≤k≤s
γk
s
n=k
|yn|+lim sup n |
xn| ,lim sup n |
xn|
≤max
sup
k≥
γk
∞
n=k
|yn|+lim sup n |xn|
,lim sup
n |xn|
=sup k∈N γk ∞
n=k
|yn|+lim sup n |xn|
.
On the other hand, let> ands∈Nsuch thaty–ys<for everys≥s. Fixs≥s.
In this case
sup
≤k≤s
γk
∞
n=k
|yn|+lim sup
n |xn| ≤
sup
≤k≤s
γk
s
n=k
|yn|++lim sup n |xn|
.
And
sup
s<k
γk
∞
n=k
|yn|+lim sup n |xn|
≤sup
s<k
γk
+lim sup
n |xn|
=+lim sup
n |xn|
In any case,
(ys)≤sup k∈N
γk
∞
n=k
|yn|+lim sup n |
xn|
≤max
sup
≤k≤s
γk
s
n=k
|yn|+lim sup n |xn|
,lim sup
n |xn|
+=(ys) +.
Taking limits whensgoes to infinity and having in mind thatis arbitrary, we obtain the
desired equality.
By using the previous equality we can now check the following.
Lemma . For the space(,| · |),w∗-null type functions are w∗-sequentially
semicon-tinuous.
Proof Let{ym}be aw∗convergent sequence and takeyitsw∗limit.
TakePkthe natural projections associated to the usual basis ofand takeQn=I–Pn.
Fixn∈N. By Lemma ., we have that(ys) =supk∈N{γk(() +Qk–(ys))} ≥γn(() +
Qn–(ys)).
It is clear thatQn–(yk) convergew∗toQn–(y). Using that · is aw∗ lower
semicon-tinuous function, we obtain that
lim inf
m→∞(ym)≥lim infm→∞γn
() +Qn–(ym)
=γn() +γnlim inf
m→∞Qn–(ym)
≥γn() +γnQn–(y).
This inequality then holds for each n∈ N, which proves that lim infm→∞(ym)≥
supn∈N{γn(() +Qn–(y))}=(y) as we wanted to prove.
Using Theorem . we finally deduce the following.
Corollary . (,| · |)has the w∗-FPP for left reversible semigroups.
Example . ConsiderX=endowed with the equivalent norm
|x|=max
x,maxn
x(n)+ ∞
n=
|x(n)| n
.
Considerτ as the weak∗topologyσ(,c). Notice that (,| · |) is a dual space since the
Schauder basis{en}is boundedly complete and monotonous. Taking the sequencexn=en
forn∈N, we obtainlimnxn= =limn,m;n=mxn–xm= , which implies thatXfails
thew∗-GGLD condition. However, it is easy to check thatXhas the Opial condition with respect to theσ(,c) topology and that theσ(,c)-null type functions are sequentially
We finish the section with the Banach space introduced in Example ., for which we showed that the asymptotic centers were not weakly compact in general, so the arguments used in [, , ] or [] are not valid for proving thew∗-FPP for left reversible semigroups.
Example . LetX:= (, · ), where the norm · is defined as
x=max
n∈N
x(n)+
∞
i=n+
x(i) .
Here x≤ x ≤ xfor everyx∈. Notice that ifx,yare vectors insuch 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 X is isometric to a dual space. Assume that {xn} is a weak∗-null sequence, where by
w∗ topology we mean theσ(,c) topology. Without loss of generality, we can assume
that {xn}is a block basic sequence,l=limnxnexists and that limn,m;n=mxn–xm=
lim supnlim supmxn–xm. Therefore
lim sup
n
lim sup
m
xn–xm=lim sup n
lim sup
m
max
xm,xn+
xm
=lim sup
n
max
l,xn+
lim supm xm
=l+
lim supm xm≥
l,
which implies thew∗-GGLD property.
Let us prove that the weak∗-null type functions arew∗-lower semicontinuous.
Take{xn}aw∗-null sequence andy=∞n=ynen∈. We can assume that{xn}is a block
basic sequence. Let> ands∈Nsuch thaty–ys< for everys≥s, whereys=
s
n=ynen. Notice that(y) =lim supny–xn=limslim supnys–xn. Moreover, from
the definition of the norm and from the equality (),
lim sup
n
ys–xn=max
lim sup
n
xn,ys+
lim supn
xn
≤max
lim sup
n xn
,y+
lim supn xn
for everys∈N. On the other hand, for everys≥s,
max
lim sup
n xn
,y+
lim supn xn
≤max
lim sup
n
xn,ys+
lim supn
xn
+
=lim sup
n
Taking limits whensgoes to infinity, we deduce that for everyy∈,
lim sup
n y
–xn=max
lim sup
n xn
,y+
lim supn xn
.
The above implies that thew∗-null type functions are constant on spheres and therefore w∗-slsc.
Therefore (, · ) verifies thew∗-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=en for n∈Nand the vector x= e. Thenlimnxn=
limnxn+x.
5 Modular function spaces and the
ρ
-FPP for left reversible semigroups In this section we consider modular function spacesLρ:={f ∈M:ρ(αf)→ asα→}endowed with the Luxemburg norm
f:=inf
α> :ρ
f
α
≤
,
and the Orlicz norm
fo:=inf
k
+ρ(kf):k>
.
HereMdenotes a set of measurable functions andρa convex additive function modular defined overM. It is said that a sequence (fn)⊂Lρconverges tof ρ-almost everywhere,
fn→f ρ-a.e., if{w∈:f(w)=limnfn(w)}isρ-null. For all general definitions we refer to
[, ] or [].
A function modularρis said to satisfy the-type condition if there exists someK>
such thatρ(f)≤Kρ(f) for everyf ∈Lρ.
In this section we assume thatρis aσ-finite convex additive function modular which satisfies the-type condition (see [] or []). In this case, a topologyτρ is defined
overLρsuch thatτρ-compact sets coincide exactly withτρ-sequentially compact sets [].
Moreover, theρ-convergence coincides with theτ-convergence up to subsequences. In [] (Section ) it is proved that, under the-type condition, the modular function
spaceLρverifies theτρ-uniform Opial condition for both the Luxemburg and the Orlicz
norm. Moreover, Lemma . and Lemma . of [] show that theτρ-null type functions areτρ-sequentially lower semicontinuous. Hence we can state the following theorem.
Theorem . Letρ be aσ-finite convex additive function modular which satisfies the
-condition.Then the modular function space Lρ verifies theτρ-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
6 L-Embedded Banach spaces and the FPP for left reversible semigroups with respect to the abstract measure topology
In this section we considerL-embedded Banach spaces endowed with the so-called mea-sure topology. A Banach spaceXis said to be anL-embedded Banach space if there exists a closed subspaceXs⊂X∗∗such thatX∗∗=X⊕Xs. A wide study and many examples of
this class of Banach spaces can be found in the monograph []. Examples ofL-embedded Banach spaces are the following:
() Duals ofM-embedded Banach spaces.
() L(μ)-spaces and preduals of von Neumann algebras.
Recall that a sequence{xn}is said to span an asymptotically isometric copy ofif there
exists a nonincreasing sequence{δn} ⊂[, ) tending to such that
∞
n=
( –δn)|αn| ≤
∞
n=
αnxn
≤
∞
n=
|αn|
for every{αn} ∈. In this case we will denotexn∼(asy).
ForL-embedded Banach spaces, the abstract measure topology (τμ) is defined in [] (Section ) by considering the class of convergent sequences. Namely, if{xn}is a sequence
in anL-embedded Banach space, we say that{xn}tends to in the abstract measure
topol-ogy (τμ-limnxn= ) if
{xn}is norm bounded and every subsequence{xnk}contains a subsequence{xnkl} such thatxnkl/xnkl ∼(asy)orxnkl →.
WhenXis a separableL-embedded Banach space, the notions of compactness and se-quential compactness agree forτμ[].
It is proved in [] (see also []) that for everyτμ-null sequence{xn}in anL-embedded
Banach space,
lim sup
n
xn+x=lim sup n
xn+x
for allx∈X. This equality implies theτμ-GGLD property and theτμ-sequential lower
semicontinuity of theτμ-null type functions.
It is known thatL-embedded Banach spaces satisfy the FPP for nonexpansive mappings with respect to the abstract measure topology []. According to Theorem ., we can extend this result to left reversible semigroups in the following way.
Corollary . Let X be a separable L-embedded Banach space.Then X verifies theτμ-FPP for left reversible semigroups.
As a particular case we can deduce Theorem . in [] when the Hilbert spaceHis sep-arable. Indeed, in case that theL-embedded Banach space isL(M,τ) for some finite von
Neumann algebraMdefined over a Hilbert space, the previous measure topology coin-cides with the usual measure topology defined onL(M,τ) for bounded sets (see The-orem . in []). Hence, noncommutativeL-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 thatL(M,τ) =L(μ) for some
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
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