In this note, we show that the order convergence in a vector lattice $X$ is
not topological unless $\dim X<\infty$. Furthermore, we show that, in atomic
order continuous Banach lattices, the order convergence is topological on order
intervals

Dabboorasad, Y. A.

Emelyanov, E. Y.

Marabeh, M. A. A.

English

arXiv

In this note, we show that the order convergence in a vector lattice is not topological unless . Furthermore, we show that, in atomic order continuous Banach lattices, the order convergence is topological on order intervals.
Subjects: Functional Analysis (math. FA)
MSC classes: 46A16, 46A40, 46B30
Cite as: arXiv: 1705.09883 [math. FA]
(or arXiv: 1705.09883 v1 [math. FA] for this version)
Submission history
From: Yousef Dabboorasad [view email]
[v1] Sun, 28 May 2017 01: 38: 39 GMT (6kb

Dabboorasad, Yousef A.m.

Emelyanov, E YU

Marabeh, Maa

Institutional Repository of the Islamic University of Gaza

arXiv:1705.09883v1  [math.FA]  28 May 2017ORDER CONVERGENCE IN INFINITE-DIMENSIONALVECTOR LATTICES IS NOT TOPOLOGICALY. A. DABBOORASAD1, E. Y. EMELYANOV1, M. A. A. MARABEH1Abstract. In this note, we show that the order convergence in a vectorlattice X is not topological unless dimX < ∞. Furthermore, we showthat, in atomic order continuous Banach lattices, the order convergenceis topological on order intervals.1. IntroductionA net (xα)α∈A in a vector lattice X is order convergent to a vector x ∈ Xif there exists a net (yβ)β∈B in X such that yβ ↓ 0 and, for each β ∈ B,there is an αβ ∈ A satisfying |xα − x| ≤ yβ for all α ≥ αβ . In this case, wewrite xαo−→ x. It should be clear that an order convergent net has an orderbounded tail. A net xα in X is said to be unbounded order convergent to avector x if, for any u ∈ X+, |xα−x|∧uo−→ 0. In this case, we say that the netxα uo-converges to x, and write xαuo−→ x. Clearly, order convergence impliesuo-convergence, and they coincide for order bounded nets. For a measurespace (Ω,Σ, µ) and a sequence fn in Lp(µ) (0 ≤ p ≤ ∞), we have fnuo−→ 0iff fn → 0 almost everywhere; see, e.g., [8, Remark 3.4]. Hence, fno−→ 0 inLp(µ) iff fn → 0 almost everywhere and fn is order bounded in Lp(µ). Itis known that almost everywhere convergence is not topological in general,i.e. there may not be a topology such that the convergence with respect tothis topology is the same as a.e.-convergence; see for example [13]. Thus,the unbounded order convergence is not topological in general.A net xα in a normed lattice X is unbounded norm convergent (un-convergent) to a vector x if, for all u ∈ X+,∥∥|xα − x| ∧ u∥∥ → 0 (cf.[11, 14, 7, 10]). In this case, we write xαun−→ x. Clearly, norm conver-gence implies un-convergence, and they agree for order bounded nets. Unlikeorder and unbounded order convergences, un-convergence is always topolog-ical, and the corresponding topology is referred to as the un-topology (see[7, Section 7]). The un-topology has been recently investigated in detail in[10].Date: 28.05.2017.2010 Mathematics Subject Classification. 46A16, 46A40, 46B30.Key words and phrases. topological convergence, vector lattice, order convergence, un-bounded order convergence, atomic vector lattice.12 Y. A. DABBOORASAD1, E. Y. EMELYANOV1, M. A. A. MARABEH1Recall that a net xα in a vector lattice X is relatively uniformly convergentto a vector x if there is u ∈ X+ such that for any ε > 0 there is αε satisfying|xα−x| ≤ εu for all α ≥ αε. In this case, we write xαru−→ x. In Archimedeanvector lattices relatively uniformly convergence implies order convergence.An element a > 0 in a vector lattice X is called an atom whenever, forevery x ∈ [0, a], there is a real λ ≥ 0 such that x = λa. It is known that theband Ba generated by an atom a is a projection band and Ba = span{a}.A vector lattice X is called atomic if the band generated by its atoms is X.For any atom a, let Pa be the band projection corresponding to Ba.A normed lattice (X, ‖·‖) is said to be order continuous if, for every netxα in X with xα ↓ 0, it holds ‖xα‖ ↓ 0 (or, equivalently, xαo−→ 0 in X implies‖xα‖ → 0). Clearly, in order continuous normed lattices, uo-convergenceimplies un-convergence.Since the order convergence could be easily not topological, many re-searchers investigated classes of ordered topological spaces, in which orderconvergence of nets (or sequences) agrees with the topological convergence.For instance, in [5], DeMarr proved that a locally convex space (X, τ) canbe made into an ordered vector space such that the convergence of nets withrespect to τ is equivalent to order convergence if and only if X is normable.In [6, Theorem 1], DeMarr showed that any locally convex space (X, τ) canbe embedded into an appropriate ordered vector space E such that xατ−→ 0iff xαuo−→ 0 in E for any net xα in X.In [15, Theorem 1], the authors characterized ordered normed spaces inwhich the order convergence of nets coincides with the norm convergence.Also, they characterized ordered normed spaces in which order convergenceand norm convergence coincide for sequences; see [15, Theorem 3].As an extension of the work in [5, 6, 15], Chuchaev investigated orderedlocally convex spaces, where the topological convergence agrees with the or-der convergence of eventually topologically bounded nets; see, for example,Theorem 2.3, Propositions 2.4, 2.5, and 2.6 in [4]. In addition, he studiedordered locally convex spaces, where the topological convergence is equiv-alent to the order convergence of sequences; see, for example, Propositions3.2, 3.3, 3.4 and 3.5 in [4].In this paper, our main result is that if (X, τ) is a topological vectorlattice such that xατ−→ 0 iff xαo−→ 0 for any net xα in X, then dimX <∞;see Theorem 1. It should be noticed that Theorem 1 was proven in thecase of ordered normed spaces with minihedral cones (see [15, Theorem2]), in the case of Banach lattices (see [16]), and in the case of normedlattices (see [9, Theorem 2]). A useful characterization of uo-convergencein atomic vector lattices is given in Proposition 1. In addition, we showthat the order convergence is topological on order intervals of atomic ordercontinuous Banach lattices; see Corollary 1. A partial converse of Corollary1 is given in Theorem 4.Throughout this paper, all vector lattices are assumed to be Archimedean.ORDER CONVERGENCE IN INFINITE-DIMENSIONAL VECTOR LATTICES IS NOT TOPOLOGICAL32. Order convergence is not topologicalIn this section, we show that order convergence is topological only infinite-dimensional vector lattices. We begin with the following technicallemma.Lemma 1. Let (X, τ) be a topological vector lattice in which xατ−→ 0 impliesxαo−→ 0 for any net xα. The following statements hold.(i) There is a strong unit e ∈ X.(ii) For any net (xα) in X, if xατ−→ 0 then xα‖·‖e−−→ 0, where ‖x‖e :=inf{λ > 0 : |x| ≤ λe}.Proof. Let N be the zero neighborhood base of τ .(i) Let ∆ := {(y, U) : U ∈ N and y ∈ U} be ordered by (y1, U1) ≤(y2, U2) iff U1 ⊇ U2. Under this order, ∆ is directed upward. Foreach α ∈ ∆, let xα = x(y,U) := y. Clearly, xατ−→ 0. Now theassumption assures that xαo−→ 0. So there are (y0, U0) ∈ ∆ and e ∈X+ such that, for all (y, U) ≥ (y0, U0), we have x(y,U) = y ∈ [−e, e].In particular, U0 ⊆ [−e, e]. Since U0 is absorbing, for every x ∈ X,there is n ∈ N satisfying |x| ∈ nU0 and so |x| ≤ ne. Hence, e is astrong unit.(ii) Since e is a strong unit then Ie = X, where Ie is the ideal generatedby e. For each x ∈ X, let ‖x‖e := inf{λ > 0 : |x| ≤ λe}. Then(X, ‖·‖e) is a normed lattice; see, for example, [3, Theorem 2.55].Suppose xατ−→ 0. Let U0 be the zero neighborhood as in part (i).Given ε > 0. Then εU0 is also a zero neighborhood. Hence, there isαε such that xα ∈ εU0 for all α ≥ αε. This implies |xα| ≤ εe for allα ≥ αε. Hence xα‖·‖e−−→ 0.Now we are ready to prove our main result, whose proof is motivated by[16].Theorem 1. Let (X, τ) be a topological vector lattice. The following state-ments are equivalent.(i) dimX <∞.(ii) xατ−→ 0 iff xαo−→ 0 for any net xα in X.Proof. The implication (i) =⇒ (ii) is trivial.(ii) =⇒ (i). It follows from Lemma 1, that X has a strong unit e, and(X = Ie, ‖·‖e) is a normed lattice. For a net xα in X, xα‖·‖e−−→ 0 ⇒ xαru−→ 0⇒ xαo−→ 0 ⇒ xατ−→ 0. Combining this with Lemma 1(ii), we get xα‖·‖e−−→ 0iff xατ−→ 0.Let (X̂, ‖·‖) be the norm completion of (X, ‖·‖e). Then (X̂, ‖·‖) is aBanach lattice. Let Îe be the ideal generated by e in X̂. Then it follows4 Y. A. DABBOORASAD1, E. Y. EMELYANOV1, M. A. A. MARABEH1from [1, Theorem 3.4], that (Îe, ‖̂·‖e) is an AM-space with a strong unit e,where ‖̂z‖e := inf{λ > 0 : |z| ≤ λe}. Now [1, Theorem 3.6] implies thatÎe is lattice isometric to C(K)-space for some compact Hausdorff space Ksuch that the strong unit e is identified with the constant function 1 on K.Clearly, X = Ie is a sublattice of Îe and so, we can identify elements of Xwith continuous functions on K.Let t0 ∈ K and g = χt0 be the characteristic function of {t0}. DefineF := {f ∈ X : f ≥ 0 and f(t0) = 1}.Then F is directed downward under the pointwise ordering. For each α ∈ F ,let fα = α. Then, Urysohn’s extension lemma assures that fα ↓ g pointwise.If g 6∈ C(K) then fα ↓ 0 in C(K), and so fα ↓ 0 in X. That is fαo−→ 0 in X,and hence fατ−→ 0 or fα‖·‖e−−→ 0, which is a contradiction since ‖fα‖e ≥ 1.Thus, g ∈ C(K), and so {t0} is open in K. So K is discrete and hence finite.Therefore, dimX <∞. Remark 1. In [15, p. 162] an example is given of an infinite-dimensionalordered normed space (which is not a normed lattice), where the norm con-vergence coincides with the order convergence.In what follows, we show that, in atomic order continuous Banach lattices,the order convergence can be topologized on order intervals. The followingresult could be known, but since we do not have an appropriate reference,we include its proof for the sake of completeness.Proposition 1. Let X be an atomic vector lattice. Then a net xα uo-converges iff it converges pointwise.Proof. Without loss of generality, we can assume that the net xα is in X+and converges to 0. The forward implication is obvious.For the converse, let xα be a pointwise null in X. Given u ∈ X+, we needto show that xα ∧ uo−→ 0. Let ∆ = Pfin(Ω) × N, where Ω is the collectionof all atoms in X. The set ∆ is directed w.r. to the following ordering:(A,n) ≤ (B,m) if A ⊆ B and n ≤ m. For each δ = (F, n) ∈ ∆, putyδ =1n∑a∈FPau +∑a∈Ω\FPau, where Pa denotes the band projection ontospan{a}. It is easy to see that yδ ↓ 0 and, for any δ ∈ ∆, there is αδ suchthat we have 0 ≤ xα ∧ u ≤ yδ for any α ≥ αδ. Therefore, xα ∧ uo−→ 0. Unlike Theorem 1, the next theorem shows that uo-convergence is topo-logical in any atomic vector lattice.Theorem 2. The uo-convergence is topological in atomic vector lattices.Proof. By Proposition 1, uo-convergence in atomic vector lattices is the sameas pointwise convergence and therefore is topological. ORDER CONVERGENCE IN INFINITE-DIMENSIONAL VECTOR LATTICES IS NOT TOPOLOGICAL5Clearly, o-convergence is nothing than eventually order bounded uo-con-vergence. Replacing “eventually order bounded” by “order bounded”, weobtain the following result in atomic vector lattices.Corollary 1. Let X be an atomic vector lattice. Then order convergence istopological on every order bounded subset of X.Proof. By Theorem 2, uo-convergence is topological in X and hence on anysubset of X in the induced topology. Since order o-convergence coincideswith uo-convergence on order intervals, we conclude that order convergenceis also topological on order bounded subsets of X. The following result extends [7, Theorem 5.3].Theorem 3. Let X be a Banach lattice. The following statements areequivalent.(i) For any net xα in X, xαuo−→ 0 ⇐⇒ xαun−→ 0.(ii) For any sequence xn in X, xnuo−→ 0 ⇐⇒ xnun−→ 0.(iii) X is order continuous and atomic.Proof. (i) =⇒ (ii) is trivial. (ii) =⇒ (iii) is part of [7, Theorem 5.3]. For (iii)=⇒ (i), suppose X is order continuous and atomic. Then, it follows from[10, Corollary 4.14], that xαun−→ 0 iff Paxα → 0 for any atom a ∈ X and, byProposition 1, this holds iff xαuo−→ 0. The following result is a partial converse of Corollary 1.Theorem 4. Assume that there is a Hausdorff locally solid topology τ onan order continuous Banach lattice X such that order convergence and τ -convergence coincide on each order interval of X. Then X is atomic.Proof. First we show that τ is a Lebesgue topology. Assume xαo−→ 0, thenthere exist α0 and ν ∈ X+ such that (xα)α≥α0 ⊆ [−ν, ν]. By the hypothesis,(xα)α≥α0τ−→ 0 in [−ν, ν], and so xατ−→ 0 in X.Let xαuo−→ 0. Since X is order continuous, then xαun−→ 0. Suppose nowxαun−→ 0, and take u ∈ X+. Then∥∥|xα| ∧ u∥∥ → 0. Since the net |xα| ∧ uis order bounded, then, by [2, Theorem 4.22], |xα| ∧ uτ−→ 0 in [−u, u], andso |xα| ∧ uo−→ 0. We conclude xαuo−→ 0. Thus xαuo−→ 0 ⇐⇒ xαun−→ 0. Itfollows from Theorem 3 that X is atomic. References[1] Y. A. Abramovich, C. D. Aliprantis, An invitation to operator theory,Vol. 50., American Mathematical Society, (2002).[2] C. D. Aliprantis, O. Burkinshaw, Locally solid Riesz spaces with appli-cations to economics, Mathematical Surveys and Monographs, Vol. 105,American Mathematical Society, (2003).[3] C. D. Aliprantis, R. Tourky, Cones and Duality, Vol. 84. Providence,RI: American Mathematical Society, (2007).6 Y. A. DABBOORASAD1, E. Y. EMELYANOV1, M. A. A. MARABEH1[4] I. I. Chuchaev, Ordered locally convex spaces in which the topologicaland order convergence coincide. Siberian Math. J., Vol. 17, no. 6,1019–1024, (1976).[5] R. DeMarr, Order convergence in linear topological spaces, Pacific J.Math., Vol. 14, 17–20, (1964).[6] R. DeMarr, Partially ordered linear spaces and locally convex linear topo-logical spaces, Illinois J. Math., Vol. 8, 601–606, (1964).[7] Y. Deng, M. O’Brien, V. G. Troitsky, Unbounded norm convergence inBanach lattices, to appear in Positivity.[8] N. Gao, V. G. Troitsky, F. Xanthos, Uo-convergence and its applicationsto Cesa´ro means in Banach lattices, to appear in Israel Journal ofMath.[9] S. G. Gorokhova, Intrinsic characterization of the space c0(A) in theclass of Banach lattices, Mathematical Notes, Vol. 60, 330–333,(1996).[10] M. Kandic´, M. A. A. Marabeh, V. G. Troitsky, Unbounded norm topol-ogy in Banach lattices, J. Math. Anal. Appl., Vol. 451, 259–279,(2017).[11] M. A. Khamsi, Ph. Turpin, Fixed points of nonexpansive mappings inBanach lattices, Proc. Amer. Math. Soc., Vol. 105, 102–110, (1989).[12] W. A. J. Luxemburg, A. C. Zaanen, Riesz spaces. Vol. I, North-HollandPublishing Co., Amsterdam-London (1971).[13] E. T. Ordman, Convergence almost everywhere is not topological,Amer. Math. Monthly, Vol. 2, 182–183, (1966).[14] V. G. Troitsky, Measures of non-compactness of operators on Banachlattices, Positivity, Vol. 8, 165-178, (2004).[15] B. Z. Vulikh, O. S. Korsakova, On spaces in which the norm convergencecoincides with the order convergence.Math. Notes, Vol. 13, no. 2, 158–163, (1973).[16] A. Wirth, Order and norm convergence in Banach lattices, GlasgowMath. J., Vol. 15, 13–13, (1974).1 Department of Mathematics, Middle East Technical University, Ankara,06800 Turkey.E-mail address: yousef.dabboorasad@metu.edu.tr, yasad@iugaza.edu.ps, ysf atef@hotmail.com,eduard@metu.edu.tr, mohammad.marabeh@metu.edu.tr, m.maraabeh@gmail.com

Order convergence in infinite-dimensional vector lattices is not topological

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Order convergence in infinite-dimensional vector lattices is not topological

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