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 Open Access
The τfixed point property for left reversible semigroups
 Francisco E CastilloSantos^{1} and
 Maria A Japón^{2}Email author
https://doi.org/10.1186/s1366301503577
© CastilloSantos and Japón 2015
 Received: 27 January 2015
 Accepted: 18 June 2015
 Published: 9 July 2015
Abstract
In this article we use the generalized GossezLami 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.
Keywords
 Schauder basis
 sequentially separating norms
 fixed point property
 nonexpansive mappings
 renorming theory
 Schur property
MSC
 46B03
 47H09
 47H10
1 Introduction
A semigroup S is said to be a semitopological semigroup if S is equipped with a Hausdorff topology such that for each \(a\in S\), the two mappings from S into S defined by \(s\to as\) and \(s\to sa\) 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 [1].
 (i)
\(ts(u)=t(su)\) for all \(t,s\in S\) and \(u\in C\).
 (ii)
For all \(u_{0}\in C\), the function \(s\in S\to s(u_{0})\in C\) is continuous.
 (iii)
For every \(s\in S\), the mapping \(u\in C\to s(u)\in C\) 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 \(\phi:S\times C\to C\), the set \(\operatorname{Fix}(S) := \{u\in C: t(u)= u,\forall t\in S\}\) is nonempty.
Definition 1.1
Let X be a Banach space and τ be a topology on X. It is said that X has the τ fixed point property (τFPP) for left reversible semigroups if every closed, convex, bounded subset C which is τcompact 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 \(\{T, T^{2}, T^{3},\ldots\}\) acting from C to C, Definition 1.1 becomes the usual definition of the τFPP for nonexpansive mappings. There exist some Banach spaces failing the wFPP [2] and therefore they fail the wFPP for left reversible semigroups (we can consider the semigroup \(S=\{T, T^{2}, T^{3},\ldots\}\) where T is the fixed point free nonexpansive mapping in the wellknown Alspach example [2]).
In 1965 Kirk proved that every Banach space with weak normal structure satisfies the wFPP 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 [3], Holmes and Lau [4] in the setting of nonexpansive actions of left reversible semigroups, that is, weak normal structure implies the wFPP 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 6.3 in [5]).
Particular examples of dual Banach spaces are known to satisfy the weak^{∗}FPP for left reversible semigroups. In 1980 Lim [6] proved that the sequence space \(\ell_{1}\) satisfies the weak^{∗}FPP for left reversible semigroups. In 2010, Lau and Mah in [5] generalized Lim’s result by proving that the FourierStieltjes 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 \(\ell_{1}(\mathbb{Z})\). In 2010, Randrianantoanina [7] proved that the space \(\mathcal {T}(H)\) 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 [8].
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 2.1.
In 2010, Randrianantoanina [7] proved that the Banach space \(L_{1}[0,1]\) or, more generally, every noncommutative \(L_{1}\)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 \(L_{1}[0,1]\)). Here the asymptotic centers of τcompact 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 τFPP for left reversible semigroups. The strict Opial condition and the generalized GossezLami 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 τFPP for left reversible semigroups. Here we will consider different types of topologies. Firstly we will regard the weak^{∗} topology in MusielakOrlicz sequence spaces, in some renormings of \(\ell_{1}\) and in some other dual Banach spaces nonisomorphic to \(\ell_{1}\). We will also consider the topology of the convergence locally in measure in some function spaces, the abstract measure topology in Lembedded Banach spaces and the topology of ρalmost 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.
2 Preliminaries
We introduce some definitions and concepts.
Notice that \(r(\cdot)\) is a continuous function for the norm topology.
In the following example we check that asymptotic centers of weak^{∗}compact sets are not weakly compact in general.
Example 2.1
Let \((T, \tau)\) be a topological space. Recall that a function \(f:T\to \mathbb{R}\) is said to be τsequentially lower semicontinuous (τslsc) if \(f(t)\le\liminf_{n} f(t_{n})\) for every sequence \((t_{n})_{n}\subset T\) with \(\tau\mbox{} \lim_{n} t_{n}=t_{0}\). From now on, let X be a Banach space and let τ be a translation invariant topology on X, that is, \(\tau\mbox{} \lim_{n} x_{n}=x\) if and only if \(\tau\mbox{} \lim_{n} (x_{n}x)=0\).
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 (1967) [9] and the generalized GossezLami Dozo property was introduced by JiménezMelado (1992) [10] when τ is the weak topology.
Definition 2.2
Definition 2.3
The τGGLD 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 2.4
Let \((X,\Vert \cdot\Vert )\) be a Banach space and τ be a topology on X. It is said that X is uniformly KadecKlee with respect to τ, \(\operatorname{UKK}(\tau)\), if for every \(\epsilon>0\) there exists some \(\delta>0\) such that whenever \(\{x_{n}\}_{n}\) is a sequence in the closed unit ball of X, which is τconvergent to a point \(x\in X\) with \(\inf_{n\ne m} \Vert x_{n}x_{m}\Vert >\epsilon\), then \(\Vert x\Vert <1\delta\).
3 Main results
Therefore, if the set C is τsequentially compact and the function \(r(\cdot)\) is τslsc, the asymptotic center \(\mathcal {AC}(\{B_{s}\}_{s\in A}, C)\) is a nonempty, τsequentially compact set. If C is convex, so is \(\mathcal{AC}(\{B_{s}\}_{s\in A}, C)\).
We now prove the following technical lemma.
Lemma 3.1
Proof
We will use a similar argument to that of Theorem 2 in [11]. Let \(\{y_{n}\}\) be a dense sequence in C and define \(\overline{y}_{n} = \sum_{i=1}^{n} \frac{y_{i}}{n}\).
Then select any \(x_{1}\in B_{s_{1}}\) such that \(\x_{1}  y_{1}\\geq r_{0} 1\).
Take \(s_{n}\in S\) such that \(r_{s_{n}} (y_{i}) \leq r_{0} + \frac{1}{n^{2}}\), \(i=1,\ldots,n\) and \(r_{s_{n}}(\overline{y}_{n})\leq r_{0} + \frac{1}{n^{2}}\). Select \(x_{n}\in B_{s_{n}}\) such that \(r_{0}  \frac{1}{n^{2}}\leq\x_{n}  \overline{y}_{n}\\).
Thus for a fixed k, it easily follows \(\lim_{n\rightarrow\infty} \x_{n}  y_{k}\= r_{0}\). Since \(\{ y_{n}\}\) is dense in C, we deduce \(\lim_{n\rightarrow\infty} \x_{n}  x\= r_{0}\) for all \(x\in C\). □
As a consequence, the following lemma is known.
Lemma 3.2
Next we obtain fixed point results by means of the τGGLD and the Opial property.
Theorem 3.3
Let X be a Banach space and τ be a topology on X. Let C be a (norm) separable closed, convex, bounded, τ compact and τsequentially compact subset of X. Let S be a left reversible semigroup generating a nonexpansive action over C. Assume that for some \(u\in C\) the previous function \(r(\cdot)\) is τslsc. If X verifies either the τGGLD property or the Opial condition with respect to τ, then \(\operatorname{Fix}(S)\neq\emptyset\). In case of the weak topology, the separability of C is not necessary.
Proof
Let \(\mathcal{F}\) be the family of nonempty, convex, τclosed and Sinvariant subsets of C. Ordering the family by inclusion and using Zorn’s lemma, we obtain a set which is minimal with respect to being nonempty, convex, τclosed and Sinvariant. We can then assume that C is the minimal set.
In case that τ coincides with the weak topology, it is known that both the wGGLD condition and the Opial condition for the weak topology imply weak normal structure [10] and therefore the wFPP for left reversible semigroups [3]. □
Notice that for separable Banach spaces and topologies τ weaker than the norm topology, the τcompactness of the domain is a superfluous assumption. Indeed, the separability of X implies that X is Lindelöf for the norm and so does for the topology τ since it is weaker that the norm topology. Thus, τsequentially compact sets are countably compact and Lindelöf, so they are τcompact.
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 3.3, the τsequential lower semicontinuity of the function \(r(\cdot)\) for some \(u\in C\) must be checked. In what follows we study equivalent and sufficient conditions to assure this statement.
Recall that given \(\{B_{s}\}_{s\in A}\) a decreasing net of bounded subsets of X we defined \(r(\cdot)\) associated to the net \(\{B_{s}\} _{s\in A}\) as \(r(x)=\lim_{s} r(x,B_{s})\). The following lemma will be very helpful to assure whether the function \(r(\cdot)\) is τsequentially lower semicontinuous.
Lemma 3.4
The function \(r(\cdot)\) is τslsc if and only if the type functions \(\Gamma(\cdot)\) are τslsc.
Proof
One implication is direct since we can take \(B_{s}=\{x_{n}: n\ge s\}\). Then \(\Gamma(x)=r(x)=\limsup_{n}\Vert x_{n}x\Vert \).
Assume that the type functions are τsequentially lower semicontinuous. Take \(\{B_{s}\}_{s\in A}\) any decreasing net of bounded subsets, and let \((y_{n})_{n}\) be a τconvergent sequence to some point \(y\in X\). We have to prove that \(r(y)\le\liminf_{n} r(y_{n})\).
Consider a sequence \((\epsilon_{n})_{n}\) of positive real numbers with \(\lim_{n}\epsilon_{n}=0\).
By (1), \(\limsup_{n}\Vert x_{n}y\Vert =r(y)\) and by (2), \(\limsup_{n}\Vert x_{n}y_{i}\Vert \le r(y_{i})\) for every \(i\in\mathbb{N}\).
In the setting of Theorem 3.3, the set C is τsequentially compact, so we can assume that the sequence \(\{x_{n}\}\) obtained in the previous lemma is τconvergent. Moreover, since the topology is translation invariant, we only need to assume that the τnull type functions are τsequentially lower semicontinuous.
Finally, we can state our main result in this section.
Theorem 3.5
Let X be a separable Banach space, τ be a translation invariant topology on X such that τcompact sets are τsequentially compact. Assume that the τnull functions are τslsc. If X verifies either the τGGLD property or the τOpial condition, X has the τFPP for left reversible semigroups.
Notice that when τ is the weak topology, the separability of X is not necessary.
4 First examples and applications for the weakstar topology
In this section we are going to apply Theorem 3.5 to several different classes of dual Banach spaces endowed with their weak^{∗} topologies.
Corollary 4.1
In case that G is a separable compact group, its FourierStieltjes algebra \(B(G)\) is the direct onesum of a sequence of finite dimensional Banach spaces (see Section 4 in [13] and Chapter I, Theorem 11.2 in [14]). Applying Corollary 4.1 we can deduce the following.
Corollary 4.2
[5] Let G be a separable compact group and \(B(G)\) be its FourierStieltjes algebra. Then \(B(G)\) satisfies the \(w^{*}\)FPP for left reversible semigroups.
We can now state an improvement of the results in [6] about the \(w^{*}\)FPP for left reversible semigroups as follows.
Theorem 4.3
Proof
When the Schauder basis is boundedly complete, the Banach space X is isomorphic to \(Z^{*}\) ([15], Proposition 1.b.4.), and we can consider the weak^{∗} topology \(\sigma(X,Z)\) where the convergence is equivalent to the pointwise convergence for bounded sequences. Now the \(w^{*}\)null type functions are \(w^{*}\)sequentially lower semicontinuous since they are constant on spheres and the \(w^{*}\)GGLD property is satisfied. □
We can apply Theorem 4.3 to more general Banach spaces. The following example is given in [16].
Example 4.4
Let \(c_{00}\) be the vector space of all sequences of scalars which are eventually zero. Denote by \([\mathbb{N}]^{< w}\) the subsets of \(\mathbb {N}\) with finite cardinality. For \(A\subset\mathbb{N}\), let \(P_{A}\) be the usual projection, that is, if \(x=(x(n))_{n}\in c_{00}\), \(P_{A}(x)\) is the vector whose coordinates are \(x(n)\) if \(n\in A\) and zero otherwise.
Corollary 4.5
[7]
Let H be a Hilbert separable Banach space. Then \(\mathcal{T}(H)\), 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 that \(\mathcal{T}(H)\) verifies the weak^{∗} uniform KadecKlee property [17] and therefore the \(w^{*}\)GGLD condition. Proposition 2.4 in [7] implies that the \(w^{*}\)null type functions are \(w^{*}\)sequentially lower semicontinuous. (Notice that in [7] the separability of the Hilbert space can be dropped.)
Corollary 4.6
[8]
The Hardy space \(H^{1}(\Delta)\) has the weak ^{∗}FPP for left reversible semigroups.
Notice that \(H^{1}(\Delta)\) is a separable Banach space, it has the \(w^{*}\)GGLD property since it verifies the \(w^{*}\)UKK condition [18], and Lemma 3.2 in [8] implies that the \(w^{*}\)null type functions are \(w^{*}\)sequentially lower semicontinuous. Here, the weak^{∗} topology refers to the isometric predual \(C(\mathbb {T})/A_{0}(\mathbb{T})\), where \(C(\mathbb{T})\) is the space of all continuous functions on the torus \(\mathbb{T}\), with the usual supremum norm, and \(A_{0}(\mathbb{T})\) is the set of boundary values of the disc algebra with zero constant term.
Recall that map \(\phi: [0,+\infty)\to[0,+\infty)\) is said to be an Orlicz function if ϕ is convex, vanishing at 0, continuous and not identically equal to zero. A sequence \(\Phi:=\{\phi_{n}\}_{n}\) of Orlicz functions is called a MusielakOrlicz function.
In case that \(\phi_{m}=\phi_{n}\) for all \(n,m\in\mathbb{N}\), we simply say that \(\ell_{\Phi}\) is an Orlicz sequence Banach space.
For Orlicz sequence Banach spaces, it is said that an Orlicz function ϕ satisfies the condition \(\delta_{2}\) if there exist some \(t_{0}>0\) and \(K>0\) such that \(\phi (2t)\le K\phi(t)\) for every \(t\in[0,t_{0}]\).
When the condition \(\delta_{2}\) is satisfied, the unit vectors form a boundedly complete normalized unconditional basis of \(\ell_{\Phi}\) ([15], Proposition 4.d.3). We denote by \(e_{n}^{*}\) the functional vector associated with \(e_{n}\) for every \(n\in\mathbb{N}\) and consider \([e_{n}^{*}]\) the closed span of the vectors \(\{e_{n}^{*}\}\) in the dual of the MusielakOrlicz space. Since the Schauder basis is monotone, the dual of the Banach space \([e_{n}^{*}]\) is isometric to \(\ell_{\phi}\) ([15], Proposition 1.b.4.). Therefore we can consider the weak^{∗} topology \(\sigma(\ell _{\Phi}, [e_{n}^{*}])\). At this point we can state the following.
Theorem 4.7
Let \(\Phi=\{\phi_{n}\}\) be a MusielakOrlicz function satisfying the condition \(\delta_{2}\). The MusielakOrlicz space satisfies the \(w^{*}\)FPP for left reversible semigroups for both the Luxemburg and the Orlicz norm.
Proof
Particular examples of MusielakOrlicz sequence spaces are the Nakano spaces \(\ell^{(p_{n})}\) where the Orlicz functions are \(\phi _{n}(t)=\vert t\vert^{p_{n}}\) with \(p_{n}\subset[1,+\infty)\) [23]. In this case, the MusielakOrlicz function satisfies the condition \(\delta_{2}\) if and only if \(\sup_{n} p_{n}<+\infty\). In case that \(p_{n}=1\) for every \(n\in\mathbb{N}\), we obtain \(\ell_{1}\). However, there exist some dual Nakano spaces which are not isomorphic to \(\ell_{1}\) and satisfying \(\lim_{n} p_{n}=1\) [23].
Corollary 4.8
Let \((p_{n})\) be a bounded sequence in \([1,+\infty)\). Then the corresponding Nakano sequence Banach space verifies the \(w^{*}\)FPP for left reversible semigroups.
At this point, we do not know whether \((\ell_{1},\!\!\cdot\!\!)\) verifies a similar statement for left reversible semigroups of nonexpansive mappings. Here we prove that \((\ell_{1},\!\!\cdot\!\!)\) does verify the \(w^{*}\)FPP for left reversible semigroups where by the weak^{∗} topology we refer to \(\sigma(\ell_{1},c_{0})\). Notice that the above norm is a dual norm, that is, if \(X=(\ell_{1},\!\!\cdot\!\!)\) then X is isometric to a dual space. This can be deduced from the fact that the Schauder basis \(\{e_{n}\}\) is boundedly complete and it is monotonous for the \(\!\!\cdot\!\!\) norm. Moreover, X is isometric to the dual of the Banach space spanned by \([e_{n}^{*}]\) so the weak^{∗} topology is in fact \(\sigma(\ell_{1},c_{0})\) (see Proposition 1.b.4 in [15]).
Lemma 4.9
Proof
We may, without loss of generality, assume that the sequence \(\{x_{n}\}\) is disjointly supported.
Let \(y= \sum_{n=1}^{\infty} y_{n} e_{n}\) be any vector in \(\ell _{1}\) and denote \(y_{s}=\sum_{n=1}^{s} y_{n} e_{n}\). Notice that Γ is a continuous function for the norm topology so \(\Gamma (y)=\lim_{s}\Gamma(y_{s})\). If we prove that \(\lim_{s}\Gamma(y_{s})\) coincides with the right part of the equality stated in the lemma, the proof will be finished.
By using the previous equality we can now check the following.
Lemma 4.10
For the space \((\ell_{1},\!\!\cdot\!\!)\), \(w^{*}\)null type functions are \(w^{*}\)sequentially semicontinuous.
Proof
Let \(\{y_{m}\}\) be a \(w^{*}\) convergent sequence and take y its \(w^{*}\) limit.
Take \(P_{k}\) the natural projections associated to the usual basis of \(\ell_{1}\) and take \(Q_{n} = I  P_{n}\). Fix \(n\in\mathbb{N}\). By Lemma 4.9, we have that \(\Gamma(y_{s}) = \sup_{k\in\mathbb{N}}\{\gamma_{k} ( \Gamma(0) + \Q_{k1} (y_{s})\_{1}) \} \geq \gamma_{n} ( \Gamma(0) + \Q_{n1} (y_{s})\_{1})\).
This inequality then holds for each \(n\in\mathbb{N}\), which proves that \(\liminf_{m\to\infty} \Gamma(y_{m})\geq\sup_{n\in\mathbb {N}}\{ \gamma_{n} ( \Gamma(0) + \Q_{n1} (y)\_{1} )\}= \Gamma(y)\) as we wanted to prove. □
Using Theorem 3.5 we finally deduce the following.
Corollary 4.11
\((\ell_{1}, \!\!\cdot\!\!)\) has the \(w^{*}\)FPP for left reversible semigroups.
Example 4.12
We finish the section with the Banach space introduced in Example 2.1, for which we showed that the asymptotic centers were not weakly compact in general, so the arguments used in [5, 6, 8] or [7] are not valid for proving the \(w^{*}\)FPP for left reversible semigroups.
Example 4.13
Let us prove that the weak^{∗}null type functions are \(w^{*}\)lower semicontinuous.
Therefore \((\ell_{1},\Vert \cdot\Vert )\) verifies the \(w^{*}\)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 \(x_{n}=e_{n}\) for \(n\in\mathbb{N}\) and the vector \(x={1\over 2}e_{1}\). Then \(\lim_{n} \Vert x_{n}\Vert = \lim_{n}\Vert x_{n}+x\Vert \).
5 Modular function spaces and the ρFPP for left reversible semigroups
A function modular ρ is said to satisfy the \(\Delta_{2}\)type condition if there exists some \(K>0\) such that \(\rho(2f)\le K \rho (f)\) for every \(f\in L_{\rho}\).
In this section we assume that ρ is a σfinite convex additive function modular which satisfies the \(\Delta_{2}\)type condition (see [27] or [28]). In this case, a topology \(\tau_{\rho}\) is defined over \(L_{\rho}\) such that \(\tau_{\rho}\)compact sets coincide exactly with \(\tau_{\rho}\)sequentially compact sets [28]. Moreover, the ρconvergence coincides with the τconvergence up to subsequences.
In [27] (Section 4) it is proved that, under the \(\Delta_{2}\)type condition, the modular function space \(L_{\rho}\) verifies the \(\tau _{\rho}\)uniform Opial condition for both the Luxemburg and the Orlicz norm. Moreover, Lemma 5.3 and Lemma 5.4 of [27] show that the \(\tau_{\rho}\)null type functions are \(\tau_{\rho}\)sequentially lower semicontinuous. Hence we can state the following theorem.
Theorem 5.1
Let ρ be a σfinite convex additive function modular which satisfies the \(\Delta_{2}\)condition. Then the modular function space \(L_{\rho}\) verifies the \(\tau_{\rho}\)FPP for left reversible semigroups when it is equipped with either the Luxemburg or the Orlicz norm.
Examples of modular function spaces are the \(L_{p}\)spaces and more generally the MusielakOrlicz function spaces, where the \(\tau_{\rho}\) topology coincides with the local convergence in measure topology. For the definition of MusielakOrlicz function spaces and for more examples of modular function spaces, see [20] and [26].
6 LEmbedded Banach spaces and the FPP for left reversible semigroups with respect to the abstract measure topology
 (1)
Duals of Membedded Banach spaces.
 (2)
\(L_{1}(\mu)\)spaces and preduals of von Neumann algebras.

\(\{x_{n}\}\) is norm bounded and every subsequence \(\{x_{n_{k}}\}\) contains a subsequence \(\{x_{n_{k_{l}}}\}\) such that \(x_{n_{k_{l}}}/\Vert x_{n_{k_{l}}}\Vert \sim(asy) \ell_{1}\) or \(\Vert x_{n_{k_{l}}}\Vert \to0\).
When X is a separable Lembedded Banach space, the notions of compactness and sequential compactness agree for \(\tau_{\mu}\) [31].
It is known that Lembedded Banach spaces satisfy the FPP for nonexpansive mappings with respect to the abstract measure topology [32]. According to Theorem 3.5, we can extend this result to left reversible semigroups in the following way.
Corollary 6.1
Let X be a separable Lembedded Banach space. Then X verifies the \(\tau_{\mu}\)FPP for left reversible semigroups.
As a particular case we can deduce Theorem 5.1 in [7] when the Hilbert space H is separable. Indeed, in case that the Lembedded Banach space is \(L^{1}(\mathcal{M},\tau)\) for some finite von Neumann algebra \(\mathcal{M}\) defined over a Hilbert space, the previous measure topology coincides with the usual measure topology defined on \(L^{1}(\mathcal{M},\tau)\) for bounded sets (see Theorem 1.1 in [31]). Hence, noncommutative \(L_{1}\)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 \(L^{1}(\mathcal{M},\tau)=L^{1}(\mu)\) for some σfinite measure space.
Declarations
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 MTM201234847C0201 and Junta de Andalucía, Grants FQM127 and P08FQM03543.
Open Access 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.
Authors’ Affiliations
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