Bibounded uo-convergence and b-property in vector lattices
aa r X i v : . [ m a t h . F A ] S e p Bibounded uo -convergence and b -property in vector lattices September 17, 2020
Safak Alpay , Eduard Emelyanov , Svetlana Gorokhova Abstract
We define bidual bounded uo -convergence in vector lattices and in-vestigate relations between this convergence and b -property. We provethat for a regular Riesz dual system h X, X ∼ i , X has b -property if andonly if the order convergence in X agrees with the order convergencein X ∼∼ . keywords: vector lattice, order dual, regular Riesz dual system, b -property, unbounded order convergence, Banach lattice MSC2020:
In the present paper, all vector lattices are supposed to be real andArchimedean. By X ∼ we denote the order dual of a vector lattice and by X ∼ n its order continuous dual. A pair h X, Y i is called a Riesz dual system if Y is an order ideal of X ∼ separating points of X [2, Def.3.51]. The natural duality in h X, Y i is h x, y i := y ( x ). Forany Riesz dual system h X, Y i , X will be identified with its imageˆ X ⊆ X ∼∼ under the canonical embedding x → i ( x ) = ˆ x , whereˆ x ( y ) := y ( x ) for y ∈ X ∼ . For a Riesz dual system h X, Y i , it is wellknown that ˆ X is a vector sublattice of Y ∼ n and hence of Y ∼ (cf. [2,p.173]). Definition 1.1.
A Riesz dual system h X, Y i is called regular if ˆ X isa regular sublattice of Y ∼ . The following proposition can be considered as a supplement toTheorem 3.54 of [2].
Proposition 1.1.
For a Riesz dual system h X, Y i , the following state-ments are equivalent. i ) h X, Y i is a regular Riesz dual system. ii ) ˆ X is a regular sublattice of Y ∼ n . iii ) Y ⊆ X ∼ n . iv ) ˆ X is an order dense sublattice of Y ∼ n .Proof. i ) = ⇒ ii ): It follows from ˆ X ⊆ Y ∼ n because ˆ X is a regularsublattice of Y ∼ in view of i ). ii ) = ⇒ iii ): Let x α ↓ X . Then ˆ x α ↓ X , and since ˆ X is a regular sublattice of Y ∼ n then also ˆ x α ↓ Y ∼ n . Hence y ( x α ) =ˆ x α ( y ) → y ∈ Y e.g. by [1, Thm.1.67]. It follows that each y ∈ Y is order continuous, as desired. iii ) ⇐⇒ iv ) is contained in Theorem 3.54 of [2]. iv ) = ⇒ ii ) is Theorem 1.23 of [1]. ii ) = ⇒ i ): Since ˆ X is a regular sublattice of Y ∼ n and Y ∼ n , being aband in Y ∼ , is a regular sublattice of Y ∼ then ˆ X is a regular sublatticeof Y ∼ . t is worth to mention that Lemma 3.2 of [8] follows directly from theequivalence iii ) ⇐⇒ vi ) of Proposition 1.1.Let X ∼ separate points of X . If X ∼ n = X ∼ then h X, X ∼ i is a regular Riesz dual system and hence ˆ X is a regular sublattice of X ∼∼ .The next fact follows now from Proposition 1.1 Corollary 1.1.
For a Riesz dual system h X, X ∼ i the following con-ditions are equivalent : i ) X ∼ n = X ∼ .ii ) ˆ X is a regular sublattice of X ∼∼ . A net x α in a vector lattice X is unbounded order convergent(briefly, uo -convergent) to x ∈ X , whenever | x α − x | ∧ u o −→ u ∈ X + . For any net x α in X , we have:( a ) x α uo −→ | x α | uo −→ b ) x α o −→ x α uo −→ x α is eventually order bounded.In particular, a functional y ∈ X ∼ belongs to X ∼ n iff y ( x α ) → x α such that x α uo −→ h X, X ∼ uo i , with X ∼ uo separat-ing points of a normed lattice X . Any functional y ∈ X ∼ which takes uo -null nets to null nets is a linear combination of the coordinate func-tionals of finitely many atoms of X ; see, e.g. [8, Prop.2.2]. Thereforethe usual way of defining the uo -dual of X fails to be interesting. Inorder to make the definition meaningful, for the case when X is anormed lattice, Gao, Leung, and Xanthos set an additional conditionon uo -null nets. Namely, they define X ∼ uo as the collection of all func-tionals from X ∼ taking norm bounded uo -null nets to null nets [8,Def.2.1]. n the case of an arbitrary vector lattice, the first candidate forsuch an additional condition, the eventually order boundedness of uo -convergent nets fails again because it turns uo –convergent nets to just o –convergent nets.Assuming X ∼ separates the points of X , we investigate anotheradditional condition, namely the eventually order boundedness of uo -null nets in X ∼∼ . Recall that a subset A of X is called b -order bounded whenever ˆ A is order bounded in X ∼∼ [3, Def.1.1]; X has b -property ,whenever every b -order bounded subset of X is order bounded [3,Def.1.1]. Definition 1.2.
Let X ∼ separate points of X . A net x α in X is called bbuo -convergent to x ∈ X if x α uo −→ x and the net ˆ x α is eventually orderbounded in X ∼∼ . Note that, in C [0 , uo -convergence, bbuo -convergence, and o -convergence agree. In particular, the bbuo -convergenceis not topological [9, Thm.2] (see also [5, Thm.2.2]).Clearly, any o -convergent net is bbuo -convergent and, by Lemma2.2, in the case when X has b -property, bbuo -convergence agrees with o -convergence. Since every order dual vector lattice X = Y ∼ has b -property, replacement of eventually b -order boundedness by eventuallyorder boundedness in 2 n -th order dual of X for some n ∈ N leads to thesame notion as the eventually b -order boundedness (the case n = 1).In the present paper we investigate relations between bbuo -convergenceand b -property in vector and Banach lattices. For further unexplainedterminology and notations we refer to [1, 2, 4, 7]. bbuo -Convergence in vector lattices In this section, we assume that X ∼ separates points of the vectorlattice X , so that h X, X ∼ i is a Riesz dual system. We begin with the ollowing two lemmas. Lemma 2.1.
For a Riesz dual system h X, X ∼ i the following condi-tions are equivalent : i ) ˆ X is a regular sublattice of X ∼∼ . ii ) x α bbuo −−−→ x implies ˆ x α o −→ ˆ x in X ∼∼ for every net x α in X and x ∈ X . iii ) X ∼ n = X ∼ .Proof. i ) = ⇒ ii ): Let x α bbuo −−−→ x . Then x α uo −→ x in X and henceˆ x α uo −→ ˆ x in X ∼∼ [7, Thm.3.2]. Since the net ˆ x α is eventually orderbounded in X ∼∼ , ˆ x α o −→ ˆ x in X ∼∼ ii ) = ⇒ i ): Let x α ↓ X . Then x α bbuo −−−→ x α o −→ X ∼∼ . Hence ˆ x α ↓ X ∼∼ . By Lemma 2.5 of [7], ˆ X is a regular sublattice of X ∼∼ . i ) ⇐⇒ iii ) is Corollary 1.1 Lemma 2.2.
Let x α be a net in a vector lattice X possessing b -property, x ∈ X . Then x α bbuo −−−→ x iff x α o −→ x .Proof. It suffices to show that x α bbuo −−−→ x α o −→
0. Let x α bbuo −−−→ X . Hence | ˆ x α | ≤ u ∈ X ∼∼ for all α ≥ α . Since X has b -property,we may assume u = ˆ w ∈ ˆ X . So, x α uo −→ X and | x α | ≤ w ∈ X forall α ≥ α . Thus x α o −→ X .We define the bbuo -dual by X ∼ bbuo := { y ∈ X ∼ | x α bbuo −−−→ ⇒ y ( x α ) → } . Since x α o −→ ⇒ x α bbuo −−−→ X ∼ bbuo ⊆ X ∼ n . Clearly, X ∼ bbuo isan order ideal in X ∼ n and hence in X ∼ . Furthermore, in the case of anormed lattice X , both X ∼ uo and X ∼ bbuo are clearly norm closed idealsin X ∼∼ , and X ∼ uo ⊆ X ∼ bbuo . We include several simple examples. Example 2.1. ( a ) ( c ) ∼ uo = ℓ , ( ℓ ) ∼ uo = c , and ( ℓ ∞ ) ∼ uo = ℓ [ , Ex. . therefore (( c ) ∼ uo ) ∼ uo = c , (( ℓ ∞ ) ∼ uo ) ∼ uo = c , and (( ℓ ) ∼ uo ) ∼ uo = ℓ . b ) ( c ) ∼ bbuo = ℓ , ( ℓ ) ∼ bbuo = ℓ ∞ , and ( ℓ ∞ ) ∼ bbuo = ℓ ; therefore (( ℓ ∞ ) ∼ bbuo ) ∼ bbuo = ℓ ∞ , (( c ) ∼ bbuo ) ∼ bbuo = ℓ ∞ , and (( ℓ ) ∼ bbuo ) ∼ bbuo = ℓ . ( c ) Let X be an atomic universally complete vector lattice, withoutlost of generality X = s (Ω) the space of real-valued functions ona set Ω . Then X ∼ bbuo = X ∼ n = X ∼ = c (Ω) the space of allreal-valued functions on Ω with finite support, and c (Ω) ∼ uo = c (Ω) ∼ bbuo = c (Ω) ∼ n = s (Ω) . Therefore ( X ∼ bbuo ) ∼ bbuo = X and (( c (Ω) ∼ bbuo ) ∼ bbuo = c (Ω) . ( d ) Let (Ω , Σ , P ) be a non-atomic probability space, ≤ p ≤ ∞ ,and L p := L p (Ω , Σ , P ) . Then ( L p ) ∼ uo = L q for < p ≤ ∞ , q − + p − = 1; and ( L ) ∼ uo = { } [ ] . On the other hand, ( L p ) ∼ bbuo = L q for all ≤ p ≤ ∞ . The following result states that X ∼ bbuo indeed coincides with X ∼ n .In particular, the duality theory for bbuo -convergence is already wellpresented in the literature. Theorem 1.
Let X ∼ n separate points of X . Then X ∼ bbuo = X ∼ n .Proof. It is enough to prove that X ∼ n ⊆ X ∼ bbuo . Let X ∋ x α bbuo −−−→ y ∈ X ∼ n . We have to show y ( x α ) →
0. Without lost of generality, weassume x α ≥ α . By Proposition 1.1, ˆ X is a regular sublatticeof ( X ∼ n ) ∼ n and hence of ( X ∼ n ) ∼ . Since ˆ x α uo −→ X then ˆ x α uo −→ X ∼ n ) ∼ by Theorem 3.2 of [7]. Take z ∈ X ∼∼ with 0 ≤ ˆ x α ≤ z for all α . Denoting by the same letters ˆ x α and z their restrictions to X ∼ n ,gives 0 ≤ ˆ x α ≤ z in ( X ∼ n ) ∼ for all α . So, the net ˆ x α is order boundedin ( X ∼ n ) ∼ and since ˆ x α uo −→ X ∼ n ) ∼ then ˆ x α o −→ X ∼ n ) ∼ . Sinceˆ y ∈ (( X ∼ n ) ∼ ) ∼ n , then ˆ y (ˆ x α ) → y ( x α ) = ˆ x α ( y ) = ˆ y (ˆ x α ) → X ∼ to separate points of X . ecall that a vector lattice X is said to be perfect if h X, X ∼ n i is aregular Riesz dual system and ˆ X = ( X ∼ n ) ∼ . By the Nakano theorem[1, Thm.3.18], the order dual X ∼ of any vector lattice is perfect. ByTheorem 1, X ∼ bbuo = X ∼ n and hence X ∼ bbuo is also perfect as a projectionband in X ∼ . Lemma 2.3.
Let h X, X ∼ i be a Riesz dual system such that for everynet x α in X : ˆ x α o −→ in X ∼∼ implies x α o −→ in X . If y α is a net in X such that ˆ y α o −→ z in X ∼∼ then y α is Cauchy in X .Proof. Let y α be a net in X satisfying ˆ y α o −→ z in X ∼∼ . Then ˆ y α isCauchy in X ∼∼ . Therefore the double net ˆ y α ′ − ˆ y α ′′ o -converges to0 in X ∼∼ . By the conditions of the lemma, y α ′ − y α ′′ o −→ X asdesired.The following result characterizes b -property in terms of bbuo -convergence. Theorem 2.
For a regular Riesz dual system h X, X ∼ i the followingconditions are equivalent : i ) X has b -property. ii ) x α bbuo −−−→ implies x α o −→ for every net x α in X.iii ) ˆ x α o −→ in X ∼∼ implies x α o −→ in X for every net x α in X.iv ) ˆ x α o −→ in X ∼∼ iff x α o −→ in X for every net x α in X. Proof. i ) = ⇒ ii ) follows from Lemma 2.2. ii ) = ⇒ iii ): If ˆ x α o −→ X ∼∼ then ˆ x α uo −→ X ∼∼ and, byregularity of ˆ X in X ∼∼ , x α uo −→ X . By the assumption, x α iseventually order bounded in X ∼∼ and hence ii ) implies x α o −→ X ,as desired. iii ) = ⇒ i ): Let X + ∋ x α ↑ and ˆ x α ≤ u ∈ X ∼∼ for all α . Weneed to show that, for some x ∈ X , there holds x α ≤ x for all α . ince X ∼∼ is Dedekind complete, ˆ X + ∋ ˆ x α ↑≤ u ∈ X ∼∼ impliesˆ x α o −→ z in X ∼∼ for some z ∈ X ∼∼ . By Lemma 2.3, x α is a Cauchynet in X . Then there exists a net y γ in X with y γ ↓ X suchthat for every γ there exists α γ satisfying | x α ′ − x α ′′ | ≤ y γ whenever α ′ , α ′′ ≥ α γ . Fix any γ and take α γ such that | x α ′ − x α ′′ | ≤ y γ forall α ′ , α ′′ ≥ α γ . In particular, x α − x α γ ≤ y γ for all α ≥ α γ andhence x α ≤ x := x α γ + y γ for all α as desired. iii ) = ⇒ iv ): In view of iii ) ⇐⇒ ii ), we need to prove that x α o −→ X implies ˆ x α o −→ X ∼∼ . This follows from regularity of ˆ X in X ∼∼ . iv ) = ⇒ iii ) is trivial.The condition that every disjoint sequence x n in X which is orderbounded in X ∼∼ is also order bounded in X does not imply the b -property. To see this, consider the first example at page 2 in [4],the Banach lattice X = ℓ ∞ ω ( T ) consisting of all countably supportedreal functions on an uncountable set T . Clearly X failed to have b -property. However X has the countable b -property in the sense of [4,p.2]. In particular, every sequence x n in X which is order bounded in X ∼∼ is also order bounded in X . bbuo -Convergence in Banach lattices In this section, we consider the Banach lattice case. We begin with thefollowing characterization of KB -spaces, which extends Proposition2.1 of [3], where the equivalence 2) ⇐⇒
3) was proved.
Theorem 3.
Let X be a Banach lattice with order continuous norm.The following conditions are equivalent. X is perfect. X is a KB -space. ) X has b -property. | x n | uo −→ w implies k x n k → for every sequence x n in X .If X ∼ uo separates points of X , then the above conditions are also equiv-alent to the following :5) X = Y ∗ for some Banach lattice Y .Proof. The implication 1) = ⇒
2) follows from the Nakano charac-terization of perfect vector lattices [2, Thm.1.71] utilizing the ordercontinuity of the norm in X .2) = ⇒ X ∼ n = X ∗ separates the points of X due to order con-tinuity of the norm in X . So, let X ∋ x α ↑ and sup α f ( x α ) < ∞ for each f ∈ ( X ∼ n ) + . Then sup α f ( x α ) < ∞ for each f ∈ X ∼ n = X ∗ .The uniform boundedness principle ensures that the set { x α } α is normbounded. Since X is a KB -space, we derive that k x α − x k → x ∈ X . Since x α ↑ and k x α − x k → x α ↑ x and hence X is perfect.2) ⇐⇒ ⇒ X be a KB -space and x n be a sequence with | x n | uo −→ w
0. Since X has order continuous norm, for each ε > x ∈ X + , there exists y ′ ∈ X ∗ + with ( | x ′ | − y ′ ) + ( x ) < ε for all x ′ ∈ X ∗ with k x ′ k ≤ ε > x n , andeach 0 ≤ x ′ , k x ′ k ≤
1, we have x ′ ( | x n | ) ≤ [ x ′ ∧ y ′ ]( | x n | ) + ( | x ′ | − y ′ ) + ( | x n | ) ≤ y ′ ( | x n | ) + ε. Therefore k x n k = k| x n |k ≤ sup { x ′ ( | x n | ) : 0 ≤ x ′ , k x ′ k ≤ } ≤ y ′ ( | x n | ) + ε. As | x n | w −→
0, lim sup k x n k ≤ ε for each ε >
0, and hence k x n k → ) = ⇒ X is order continuous, for provingthat X is a KB -space, it is enough to show that k x n k → x n satisfying 0 ≤ | x n | ≤ x ′′ for some x ′′ ∈ X ∗∗ . Leta sequence x n in X be disjoint and 0 ≤ | x n | ≤ x ′′ ∈ X ∗∗ for all n . Foreach x ′ ∈ X ∗ + , we have m X n =1 x ′ ( | x n | ) = x ′ m X n =1 | x n | ! = x ′ m _ n =1 | x n | ! ≤ x ′ ( x ′′ ) , and hence ∞ P n =1 x ′ ( | x n | ) < ∞ . Therefore x n w −→
0. Since each disjointsequence in X is uo -null, it follows from 4) that k x n k → ⇒ X = Y ∗ = Y ∼ . Note thatproving this implication we did not use that X ∼ uo separates points of X . 2) = ⇒ X ∼ uo is the Banachlattice ( X ∼ n ) a that is the order continuous part of X ∼ n . Since X is a KB -space, X is monotonically complete. Applying Theorem 3.4 of[8] gives that X is lattice isomorphic to the dual space ( X ∼ uo ) ∗ under i ( x )( y ) = y ( x ) for x ∈ X , y ∈ X ∼ uo . Since both X and ( X ∼ uo ) ∗ areBanach lattices, the bijection i : X → ( X ∼ uo ) ∗ is also a homeomor-phism. As it was pointed out in [8] after the proof of [8, Thm.3.4], i is an isometry iff the closed unit ball B X is order closed. The later isclearly true since X is a KB -space. So, X is lattice isometric to thedual space ( X ∼ uo ) ∗ .Notice that the condition 4) of Theorem 3 cannot be replaced by:4 ′ ) for every sequence x n in X , x n uo −→ w k x n k → ′ ) is equivalent tothe positive Schur property, which is, in general, stronger than KB . learly X ∼ uo separates points of X if the Banach lattice X is atomic.Another case when X ∼ uo separates points of X is described in Lemma2.2 of [10]. Taking these two cases together, we get immediately fromProposition 3 the following characterization. Corollary 3.1.
Let X be a Banach lattice with order continuousnorm. If X is either atomic or else a rearrangement invariant spaceon a non-atomic probability space such that X is not an AL -space thenthe following conditions are equivalent. X is perfect. X is a KB -space. X has b -property. X = Y ∗ for some Banach lattice Y .Furthermore, in this case, X is lattice isometric to ( X ∼ uo ) ∗ . The following result is similar to Theorem 2.3 of [8], that charac-terizes the dual of X ∼ uo . Unlike in Theorem 1, X ∼ n is not required tobe separating points of X . Theorem 4.
Let y be an order continuous functional on a Banachlattice X . The following conditions are equivalent :1) y ∈ X ∼ bbuo . y ( x n ) → for each b -bounded uo -null sequence x n in X . y ( x n ) → for each b -bounded disjoint sequence in X .Proof.
1) = ⇒
2) = ⇒
3) are clear.3) = ⇒ y ∈ X ∼ n satisfies y ( x n ) → b -bounded disjoint sequence in X . Let x α be an eventually b -boundedand uo -null net in X . We show y ( x α ) →
0. Without lost of generality,we assume the net x α to be b -bounded itself, say − z ≤ ˆ x α ≤ z ∈ ∼∼ . Let A be the solid hull of [ − z, z ] ∩ ˆ X in X ∼∼ . Clearly, A ⊆ [ − z, z ]. Each disjoint sequence in A is a disjoint sequence in [ − z, z ] andtherefore weakly converges to zero. So we see that, for each disjointsequence x n in A , | y | ( x n ) →
0. Now applying this observation toTheorem 4.36 of [2] for the norm continuous seminorm p ( x ) = | y | ( | x | ),we see that, for ε >
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