On the dual positive Schur property in Banach lattices

The paper contains several characterizations of Banach lattices $$E$$ with the dual positive Schur property (i.e., $$0 \le f_n \xrightarrow {\sigma (E^*,E)} 0$$ implies $$\Vert f_n\Vert \rightarrow 0$$) and various examples of spaces having this property. We also investigate relationships between the dual positive Schur property, the positive Schur property, the positive Grothendieck property and the weak Dunford–Pettis property.


Introduction
Several types of the Schur property is considered in the theory of Banach lattices. Namely, a Banach lattice E = (E, · ) has -the Schur property, if x n → 0 weakly, then x n → 0, -the positive Schur property, whenever 0 x n → 0 weakly implies x n → 0, -the strong Schur property if there exists a number K > 0 such that for all δ ∈ (0, 2] every δ-separated sequence in the unit ball contains a subsequence K δ-equivalent to the standard 1 -basis, -the dual positive Schur property, if 0 f n → 0 in the weak * topology, then f n → 0. Results concerning the first three Schur type properties are scattered in a literature but the most essential characterizations, descriptions, examples etc. one can find in [4,10,16,25,30,33]. The present paper is devoted to the fourth property which was introduced in [6]. We present several characterizations of the property (Sect. 2) and examples of Banach lattices having the dual positive Schur property (Sect. 4). Our terminology and notations are standard-we follow by [2,3,23,38].
Investigating the dual positive Schur property we easily find its relationships with the Grothendieck and the positive Grothendieck properties. Let us recall that a Banach lattice E has -the Grothendieck property if weak * null sequences in E * are weak null, -the positive Grothendieck property if weak * null sequences with positive terms are weak null.
Results concerning the Grothendieck property are presented in [23,Section 5.3] and in [26] (see also [12] where classical function spaces having the Grothendieck property are investigated).
The lattice operations are discontinuous in the weak and weak * topologies in infinite dimensional spaces (see [2,Theorem 2.36]) which is a reason that "positive" properties are weaker than their "non-positive" prototypes. Sometimes the modulus is weakly sequentially continuous or sequentially weak * continuous. Indeed, since the weak convergence of sequences in spaces C(K ) means their norm boundedness and pointwise convergence then the modulus is weakly sequentially continuous in C(K ). The same holds in discrete Banach lattices with order continuous norms (see [23,Proposition 2.5.23]). Moreover, as it was noticed in [32, Theorem 6.6], the modulus is sequentially weak * continuous in the dual of a σ -Dedekind complete Banach lattice E if and only if E is discrete and its norm is order continuous. Below we present an alternative more elementary proof, based on the Schur property, of "if" part. Proof Let (e α ) α∈A be a complete disjoint system in E consisting of discrete elements. Since the norm in E is order continuous, then every x ∈ E is of the form x = α t α (x)e α where numbers t α (x) are uniquely determined and {α : t α (x) = 0} is at most countable. Functionals f α defined by f α (x) = t α (x) are homomorphisms, and so they are discrete in E * . It is easy to see that ( f α ) α∈A is a complete disjoint system in E * . According to [2,Theorem 1.75] for an arbitrary f ∈ E * there exists a unique family (s α ( f )) α∈A of numbers such that f = (o) α s α ( f ) f α , i.e. there exists a net (g ) ⊂ E * + decreasing to zero, indexed by finite nonempty subsets of A, Suppose g n → 0 in the weak * topology and fix x = α t α (x)e α ∈ E. Since the convergence is unconditional then for every a = (a α ) and we can restrict our considerations to the band B where (e α ) α∈A 0 is an unconditional basis.
Fix f ∈ E * + . Since the expression of every x ∈ E is unconditional, then the equality f a (x) = n a n t n (x) f (e n ) defines a continuous linear functional f a on E for every a = (a n ) ∈ ∞ . Therefore (t n (x) f (e n )) ∈ 1 . Hence elements c m = (t n (x m ) f (e n )) form a weak null sequence in 1 because n a n t n (x m ) f (e n ) = f a (x m ) → 0 for every a = (a n ) ∈ ∞ . Using the Schur property we obtain f (|x m |) = n |t n (x m )| f (e n ) → 0, i.e., |x m | → 0 in the weak topology. Remark 1.2 It is worth to recall two results due to Chen and Wickstead related to Proposition 1.1.
1. If E is a Banach lattice without discrete elements and the norm on E is order continuous, then for every x ∈ E + there exists a weak null sequence (x n ) with |x n | = x for every n (see [8,Theorem 2.2]). 2. If the dual E * does not contain discrete elements, then for every f ∈ E * + there exists weak * null sequence ( f n ) such that | f n | = f for every n (see [9,Theorem 3.1]).
It is easy to observe that the modulus preserves weak convergence of sequences with disjoint terms. Using [1, Lemma 1.6.1] and an idea from [31] we obtain the following result. Proof Sequences (x n ), (|x n |) form unconditional basic sequences and x ∈ span{x n : n ∈ N}. If ( f n ) is the sequence of biorthogonal functionals associated to (x n ) then f k (x) = lim n→∞ f k (x n ) = 0, and so x = 0. The equality | m k=n t k |x k || = | m k=n t k x k | shows that the basic sequences are equivalent. Therefore span{x n : n ∈ N} and span{|x n | : n ∈ N} are isomorphic and there exists an isomorphism mapping x n onto |x n | (see [20] a remark after Definition 1.a.7). Finally |x n | → 0 weakly because the weak topology of a subspace is a restriction of the weak topology to this subspace.
An analogue of Proposition 1.3 is false for weak * convergence. An example presented below is based on an idea borrowed from [9]. Let K be a compact space containing a sequence (s n ) convergent to s 0 such that s k = s m for k = m. Denote by e n the evaluation at point s n . Define F n ∈ C(K ) * putting F n = e 3n−3 − e 3n−2 + e 3n−1 − e 3n for even n and F n = e 3n−2 − e 3n−1 for odd n. There holds F m ⊥ F j for m = j and F n → 0 * -weakly. On the other hand |F 2n | → 4e 0 and |F 2n−1 | → 2e 0 in the weak * topology. Since e 0 = 0, then the weak * limit of (|F n |) does not exists.
Our example together with next proposition gives an alternative proof of the following known fact.
Let K be an infinite quasi-Stonian compact space. A sequence (s n ) ⊂ K is convergent if and only if it is eventually constant.

Proposition 1.4 Suppose that a Banach lattice E is a σ -Dedekind complete.
Proof Let a sequence ( f n ) consisting of disjoint elements be weak * convergent to f . Choose x ∈ E and ε > 0. According to [3,Theorem 4.42] we can find g ∈ E * + satisfying the condition (| f n | − g) + (|x|) ε 2 for all n. Put h n = (| f n | − g) + .
Since | f n | h n + g, then by Riesz decomposition property | f n | = u n + v n where u n , v n ∈ E * + and u n h n , v n g.
Clearly v n | f n | and so v m ∧ v k = 0. Hence m n=1 v n (|x|) = ( m n=1 v n )(|x|) g(|x|) and we obtain v n (|x|) < ε 2 for sufficiently large n. Therefore || f n |(x)| | f n |(|x|) = u n (|x|) + v n (|x|) h n (|x|) + v n (|x|) < ε for large n. We have just proved | f n | → 0 in the weak * topology what implies Remark 1.5 As we showed before, Proposition 1.4 is false if we reject the assumption of the σ -Dedekind completeness. On the other hand there exist non σ -Dedekind complete Banach lattices for which the conclusion of Proposition 1.4 remains true. Indeed, it is well known that ∞ ∈ (GP) and that the Grothendieck property is preserved by quotients. Hence the conclusion of Proposition 1.4 holds for E = ∞ /c 0 due to Proposition 1.3 but ∞ /c 0 is not σ -Dedekind complete. The last equivalence has several simple consequences.
Proof (a) The positive Schur property implies the order continuity of a norm.
(b) It is enough to apply the following characterization: a norm in E is order continuous if and only if norm bounded disjoint positive functionals on E are weak * convergent to zero (see [32,Theorem 2.2]).
(c) is obvious as in immediate consequence of (a) and (b).
(d) If E were separable then the norm on E would be order continuous which contradicts (b). Similarly, supposing E * ∈ (SP) we obtain E does not contain any subspace isomorphic to 1 (see [30,Theorem 1]). According to [32,Theorem 4.7] both E and E * have order continuous norms which is impossible by (b). The dual E * can not be discrete because for discrete spaces the (PSP) and (SP) coincide.
A result of Dodds-Fremlin ([11, Corollary 2.7]) will be useful for our further considerations. We remind the reader of it below (an elegant and short proof may be found in [13,Proposition 4]).

Theorem 2.2 Let E be a Banach lattice. A sequence
Our next result shows that we can restrict sequences appearing in the definition of the dual positive Schur property to disjoint sequences. It is useful when we want to verify if a concrete Banach lattice has the dual positive Schur property.

Proposition 2.3 A Banach lattice E has the dual positive Schur property if and only
if an arbitrary sequence of positive disjoint functionals weak * convergent to zero is norm null.
Proof Let E * + f n → 0 in the weak * topology and let disjoint terms of a sequence (x n ) belong to positive part of the unit ball of E. According to Theorem 2.2 it is sufficient to show f n (x n ) → 0. Suppose that our conjecture is false. Passing to a subsequence if necessary we can assume f n (x n ) > ε > 0 for all n. Applying [32, Proposition 0.3.11] we find g n ∈ E * + such that g n ∧ g m = 0, g n f n and g n (x n ) = f n (x n ). Therefore (g n ) is weak * null, and it should be g n → 0 but ε < f n (x n ) = g n (x n ) g n , a contradiction.
It is worth to note that the condition does not characterize the positive Grothendieck property. Indeed, every AL-space E satisfy the condition (+) because weak * convergent sequences ( f n ) in E * are order bounded, and so they are weakly convergent whenever f k ∧ f m = 0. On the other hand none infinite dimensional AL-space E ∈ (PGP). If it were true that E ∈ (PGP) there would be 1 ∈ (PGP) because 1 is positively complemented in E but for functionals f n ∈ ( 1 ) * represented by the characteristic functions of sets N {1, . . . , n} there holds f n → 0 in the weak * topology and F( f n ) = 1, where F is the Banach limit, i.e., ( f n ) is not weak null. The authors of [6] noticed that the dual positive Schur property is related to the so called bi-sequence property. We say that a Banach lattice E satisfies the bi-sequence property if f n (x n ) → 0 for every weak null sequence (x n ) ⊂ E + consisting of disjoint elements and every weak * null sequence ( f n ) ⊂ E * + . Our succeeding result shows conditions equivalent to the bi-sequence property.

Proposition 2.4 For a Banach lattice E the following statements are equivalent. (a) For arbitrary sequences
be weak null and weak * null respectively. Since X = {x n : n ∈ N} ∪ {0} is weakly compact, then according to [3,Theorem 4.34] v n → 0 weakly for every sequence (v n ) ⊂ sol X consisting of disjoint elements. The bi-sequence property implies sup Below we formulate a few simple characterizations of the dual positive Schur property. They show the property is related to the weak Dunford-Pettis property considered in [26,Section 9], [5,31]. Let us recall that a Banach lattice E has the weak Dunford-Pettis property (shortly E ∈ (wDPP)) if every weakly compact operator on E maps weak null disjoint sequences onto norm convergent. As it was shown in [5, Theorem 2.2] the condition E ∈ (wDPP) means exactly that weakly compact operators on E transform weak null positive sequences onto norm convergent. Let us note that σ -Dedekind complete Banach lattices E satisfying ( * ) are finite dimensional. Indeed, condition ( * ) implies E does not contain any order isomorphic copy of ∞ . On the contrary there would exist a positive projection P : E → ∞ . On account of [32, Theorem 6.5] we can choose a positive non compact operator T : ∞ → L 1 [0, 1]. Thus the composition T •P is a non compact map from E into L 1 [0, 1], a contradiction. Since E is σ -Dedekind complete and contains no copies of ∞ its norm is order continuous. Condition ( * ) implies the identity on E is compact, i.e., E is finite dimensional.
On the other hand there exist spaces C(K ) satisfying ( * ) and we can characterize them.

Proposition 2.8
For an AM-space E with an order unit the following statements are equivalent.
(a) E * is order isomorphic to 1 ( ) for some set . (b) ⇒ (c) Suppose F is a Banach lattice with the order continuous norm. Every positive operator T : E → F maps the unit ball, which is an order bounded set, into a weakly compact set in F because order intervals in F are weakly compact. The space E has the Dunford-Pettis property, and so T is a Dunford-Pettis operator but classes of Dunford-Pettis operators and compact operators defined on spaces without copies of 1 are the same.

Remark 4.5.3]).
We should also mention a necessary condition for the dual positive Schur property which is due to Wójtowicz (see [35,Theorem 1]). Since the result is presented in [35] without proof we show it below with details.

Proposition 2.10 If a Banach lattice E has the dual positive Schur property, then E can not be positively and surjectively mapped onto any infinite dimensional Banach lattice with order continuous norm.
Proof Assume that an infinite dimensional Banach lattice F possesses order continuous norm and T : E → F is a positive surjection. The adjoint operator T * is a positive isomorphic embedding by the closed range theorem (see [27,). Since F * is infinite dimensional it contains a sequence (g n ) of norm one positive disjoint element. It has to be g n → 0 in the weak * topology according to the order continuity of F (see [32,Theorem 2.2]). Clearly T * (g n ) = 0 by injectivity of T * . Put f n = T * (g n )/ T * (g n ) . There holds 0 f n , f n = 1 and M = inf T * (g n ) > 0. Hence | f n (x)| M −1 |g n (T (x))| for every x ∈ E, i.e., f n → 0 in the weak * topology, which contradicts the dual positive Schur property of E.
In the last section we will show that the theorem converse to Proposition 2.10 is false. We finish this part formulating some characterization of the positive Grothendieck property which is closely related to exercise 5.3.E2 in [23]. The following version of Räbiger's Lemma (see [26, Lemma 2.1]) will be useful as a tool of a proof. We omit the proof of Lemma 2.11 because it is enough to repeat arguments used in [26]. Proof (a) ⇒ (b) Suppose F is a non reflexive Banach lattice whose norm is order continuous. The space F contains an order isomorphic copy of 1 or c 0 (see [ − −−−− → 0. According to Lemma 2.11 and the Rosenthal's 1 -theorem the sequence ( f n ) contains a weak Cauchy subsequence ( f n k ), and so ( f n k ) is weakly convergent. Hence f n → 0 in the weak topology.

A remark on λ-sums of Banach lattices
Let E k = (E k , · ), k ∈ N, be Banach lattices and let λ = (λ, · λ ) be a Banach lattice which is an ideal in R N containing all unit vectors. The space (⊕E k ) λ , called the λ-sum of spaces E k , is defined as follows The space (⊕E k ) λ is a Banach lattice under the order induced from the product k E k and under the norm (x k ) = ( x k k ) λ . It is well known that the space λ n of order continuous functionals on λ (which are norm continuous because they are order bounded-see [3, Lemma 1.54]) can be identified with the Köthe dual ((a k )) = ∞ k=1 a k b k and λ × is considered with the norm (b k ) = sup (a k ) λ 1 | k a k b k |. The space λ × becomes a Banach lattice under the standard pointwise order. Since λ × contains unit vectors and it is complete then λ × contains sequences with strictly positive terms. For more details concerning Köthe duals we refer the reader to [17, Chapter 6, section 1] and [37,Chapter 15].
A description of the space of order continuous functionals on (⊕E k ) λ is an aim of our present considerations. We will need an additional notation. Namely, if a sequence ( n k=1 x k ) is order convergent to x we will write (0, . . . , 0, x, 0, . . . ) where x is located on k-th place. Clearly J k is an order isomorphism onto a band in E. The spaces of order continuous functionals on E and E k will be denoted by E n and (E k ) n respectively.
Fix f ∈ E n and put Let us note that for every 0 The last observation implies injectivity of T . Moreover T is a surjection. Indeed, fix ( f k ) ∈ (⊕(E k ) n ) λ × . For an arbitrary x = (x k ) ∈ E the series ∞ k=1 f k (x k ) is absolutely convergent and the formula f (x) = ∞ k=1 f k (x k ) defines a linear functional on E. We claim f ∈ E n . Assume x α = (x α (k)) ↓ α 0 in E. Choose ε > 0 and α 0 . Let k 0 be such that ∞ k=k 0 +1 | f k (x α 0 (k))| ε 2 . Since x α (k) ↓ 0 in E k and f k ∈ (E k ) n , then k 0 k=1 | f k (x α (k))| < ε 2 for sufficiently large α α 0 . Hence | f (x α )| < ε for large α.
Our previous considerations lead us to the following result.

Proposition 3.1 If E = (⊕E k ) λ , then the space E n of order continuous functionals on E is order isomorphic (and so linearly homeomorphic
The λ-sums allows us to produce examples of Banach lattices with various properties. For instance repeating arguments used in [30] we can prove that if λ ∈ (SP), then (⊕E k ) λ ∈ (SP) whenever all E k have the Schur property and (⊕E k ) λ ∈ (PSP) when E k ∈ (PSP) for all k.

Examples
It is easy to check that C(K ) ∈ (DPSP) (compare [6]) for every compact space K because positive functionals on C(K ) attend their norms on the constant function equals one. Using [23,Theorem 5.3.13] and remembering that (C(K )) * ∈ (PSP), we obtain quickly a characterization of AM-spaces having the positive dual Schur property. If an AM-space E ∈ (DPSP), then also its even duals have the dual positive Schur property because they are AM-spaces with a strong unit. Below we show that many Musielak-Orlicz spaces have the dual positive Schur property.
Modifying ideas used by Leung in [19] we can prove the following result.
Proof Let 0 f n → 0 in the weak * topology and suppose f n 0. Passing to a subsequence, if necessary, and using Theorem 2.2 we can assume f n (x n ) > 2η > 0 for some norm bounded disjoint sequence (x n ) ⊂ L ψ (μ) + . Denote M = sup n x n and put A n,k = {t : x n (t) > k}. Since 1 T ∈ L ψ (μ) then 2η < f n (x n 1 A n,k ) + k f n (1 T ) and the weak * convergence of ( f n ) allows us to find a subsequence (n k ) such that η < f n k (x n k 1 A n k ,k ). Define functions y k = 1 2(M+1) x n k 1 A n k ,k . For an arbitrary ε > 0 there exists r 0 having a property ψ( 1 2 r, t) < εψ(r, t) for all t ∈ T and r r 0 . Taking , t) dμ, and so the sequence ( m j=1 y k j ) is norm bounded in L ψ (μ). According to the Levi property sup j y k j = y exists in L ψ (μ) but f n k j (y) f n k j (y n k j ) > η(2(M + 1)) −1 > 0, a contradiction because f n k j → 0 in the weak * topology.
Let us note that if L ψ (μ) ns denotes the space of continuous singular functionals on L ψ (μ) (i.e., the orthogonal complement of L ψ (μ) n ), then L ψ (μ) ns = { f ∈ (L ψ (μ)) * : f (L ψ (μ) A ) = {0}}. Indeed, as we said before L ψ (μ) A is order dense in L ψ (μ), and so a functional f vanishing on L ψ (μ) A has the trivial carrier. Using [39,Theorem 24.2] we obtain f ∈ L ψ (μ) ns . Conversely, if f ∈ L ψ (μ) ns and additionally the carrier of f reduces to zero, then the absolute kernel N f of f is order dense. Therefore every 0 x ∈ L ψ (μ) A is a supremum of an increasing net of positive elements (x α ) ⊂ N f . Since L ψ (μ) A is an ideal, then x α ∈ L ψ (μ) A and x α → x in norm. We obtained that f vanishes on L ψ (μ) A . Now consider a functional f ∈ L ψ (μ) ns whose carrier is nonzero. By [39,Theorem 24.3] | f | = sup α f α for some increasing net ( f α ) of singular functionals with trivial carriers. There holds As a consequence of our previous considerations we obtain that L ψ (μ) ns is order isomorphic and isometric to (L ψ (μ)/L ψ (μ) A ) * but the quotient L ψ (μ)/L ψ (μ) A is an AM-space (see [29] the proof of Theorem 10). Finally L ψ (μ) ns is an AL-space, and we get the following result: if L ψ (μ) ∈ (DPSP), then all even duals of L ψ (μ) have the dual positive Schur property. Indeed, Musielak-Orlicz spaces L ψ (μ) having the dual positive Schur property are perfect because L ψ (μ) n = L ψ * (μ) has an order continuous norm as a band in the space (L ψ (μ)) * ∈ (PSP). Therefore (L ψ * (μ)) * = L ψ * (μ) n = L ψ * * (μ) = L ψ (μ). Hence even duals of L ψ (μ) are of the form L ψ (μ) ⊕ C(K ) for some compact space K . Arguments applied above allow us to formulate more general result.
If a Banach lattice E ∈ (DPSP) is perfect and E is a semi-M-space (i.e., E ns is order isomorphic and isometric to an AL-space ), then even duals of E have the dual positive Schur property.

Proposition 4.3
For a Musielak-Orlicz space L ψ (μ) the following statements are equivalent.
Considering our next example we will need the notion of the Riesz angle, denoted by μ 2 (E), which plays an important role in the fixed point theory (see [7]). For a Banach lattice E the Riesz angle is defined by A behavior of μ 2 (E) is related to semi-M-spaces. The following result was proved in [34]. Consider the space E = (⊕ q k k ) ∞ where 1 < q k → ∞. E is a semi-M-space because μ 2 ( q k k ) = 2 1 q k . According to Proposition 3.1 we have E n = (⊕ p k k ) 1 (where 1 p k + 1 q k = 1). Therefore E n ∈ (SP) (see [30]). Since E * = E n ⊕ L 1 (ν) for some measure ν, then E * ∈ (PSP). Räbiger showed in [26,Satz 9.7] that E ∈ (GP). Hence E ∈ (DPSP) by Proposition 2.5 and also even duals of E have the dual positive Schur property because E is perfect.
All our examples of Banach lattices with the dual positive Schur property are such that their even duals have this property too. It is natural to ask if there exists a Banach lattice E ∈ (DPSP) with E * * / ∈ (DPSP). We finish the paper presenting an example of a Banach lattice which confirms that the theorem converse to Proposition 2.10 is false. Hernadez in [15] indicated a hereditarily 1 Orlicz sequence space ϕ (i.e., every infinite dimensional closed subspace of ϕ contains an isomorphic copy of 1 ) without the Schur property. The space ϕ has order continuous norm because it does not contain c 0 . Put E = ϕ * . Since E * is order isomorphic to ϕ ⊕ L 1 (ν) then it is weakly sequentially complete. Applying [18,Theorem 6] (or [12, Proposition 5.2]) we obtain E ∈ (GP). Moreover E / ∈ (DPSP). Indeed, if there were E ∈ (DPSP), then E * would have the positive Schur property. Hence ϕ ∈ (PSP) but discreteness of ϕ implies ϕ ∈ (SP), a contradiction.
We claim that E can not be positively mapped onto any Banach lattice with order continuous norm. On the contrary, suppose that there exists a Banach lattice F whose norm is order continuous and T : E → F is a positive surjection. Repeating arguments used in the first part of the proof of Proposition 2.12 we obtain F has to be reflexive. By the closed range theorem the adjoint operator T * : F * → E * = ϕ ⊕L 1 (ν) is a positive isomorphic embedding. Let P : E * → ϕ and Q : E * → L 1 (ν) be band projections. According to the reflexivity of F * every norm bounded sequence ( f n ) ⊂ F * contains weakly convergent subsequences ( f + n k ), ( f − n k ). Therefore (Q •T * ( f + n k )), (Q •T * ( f − n k )) are weakly convergent positive sequences in L 1 (ν), and so they are norm convergent because L 1 (ν) ∈ (PSP). We have just shown that Q • T * is compact.
The subspace (P • T * )(F * ) is closed in ϕ and it has to be finite dimensional because (P • T * )(F * ) is reflexive as a continuous image of a reflexive space and ϕ is hereditarily 1 . Hence P • T * is compact. Finally T * is compact being the sum of two compact operators which contradicts the fact that T * is an isomorphism.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.