The Köthe Dual of an Abstract Banach Lattice

We analyze a suitable definition of Köthe dual for spaces of integrable functions with respect to vector measures defined on δ-rings. This family represents a broad class of Banach lattices, and nowadays it seems to be the biggest class of spaces supported by integral structures, that is, the largest class in which an integral representation of some elements of the dual makes sense. In order to check the appropriateness of our definition, we analyze how far the coincidence of the Köthe dual with the topological dual is preserved.


Introduction
The aim of this paper is to extend the usual notion of Köthe dual of a Banach function space over a -finite measure to the general class of Banach lattices.Recall that the Köthe dual or associate space of a Banach function space  over a -finite measure  is defined by the set of all the functionals belonging to the topological dual  * that can be represented as integrals; that is, if  ∈  * is such a functional, there is a measurable function   such that   ∫    equals () for all  ∈ .
Integration with respect to Banach space valued vector measures on -rings is at the basis of a fruitful representation technique for abstract lattices that has been recently developed.Certainly, the spaces  1 (]) of integrable functions and  1   (]) of weakly integrable functions represent a large family of Banach lattices.Nowadays it is well known that each order continuous Banach lattice can be written (isometrically and in order) as an  1 (])-space of a certain vector measure ] on a -ring, and an equivalent result holds for Banach lattices with the Fatou property and some additional requirements with the spaces  1   (]) (see [1,Theorems 4 and 8] and [2]; see also [3, pp. 22-23]).Similar results follow also under the assumption of -convexity of the Banach lattices, using in this case spaces of -integrable functions (see [4][5][6]).
Far from being a formal requirement, the use of vector measures on -rings is essential for our purposes.The reason is that this is the way of extending the classical representation results-that hold for Banach function spaces based on finite measure spaces-to the abstract case, that include for instance representations of spaces ℓ  (Γ) for an uncountable set of indexes Γ based on an integration structure.On the other hand, it must be said that the representation for the finite case is completely covered by the integration structure provided by vector measures on -algebras.Regarding spaces of integrable functions with respect to such vector measures, several descriptions of the dual space of an  1 -space (an  space) of a vector measure are nowadays known (see [7][8][9][10]).Notice that, since the spaces are order continuous, these results give directly a description of the corresponding dual spaces.
In this paper we propose a concrete definition of Köthe dual space for a general class of Banach lattices.Our purpose includes two natural properties that one can expect for such a space; that is, (1) its elements must be-at least locally-integrals, that is, functionals defined by integrable functions, and (2) the set of such elements (the Köthe dual) coincides with the whole topological dual whenever the original lattice is order continuous.
As the reader may notice, we will give an easy description of such a space which covers the hole existing in

Preliminaries
2.1.Banach Lattices.We recall in this section the notions concerning Banach lattices required for a suitable reading of the paper.
Let  be a Banach lattice with order ≤ and norm ‖ ⋅ ‖.A linear subspace  of  is called an ideal of  if  ∈ , || ≤ || implies  ∈ .An ideal  in  is said to be super order dense in  if for every 0 ≤  ∈  there exists an increasing sequence (  )  ⊂  so that 0 ≤   ↑  in .If this property holds by means of upwards directed systems,  is said to be order dense in .An element 0 ≤  ∈  such that  ∧  = 0 implies  = 0 is said to be a weak unit of .
The Banach lattice  is said to have -order continuous norm, or briefly, to be -order continuous if for every decreasing sequence (  )  ↓ 0 it follows that ‖  ‖ ↓ 0. Similarly,  is said to have order continuous norm, or briefly, to be order continuous if this property holds for downwards directed systems.We denote by   the -order continuous part of , that is, the largest -order continuous ideal in .We will use   for the order continuous part of  which in this case is the largest order continuous ideal in .
Finally, the Banach lattice  is said to have the -Fatou property if for every increasing sequence 0 ≤   ↑ in  with sup  ‖  ‖ < ∞, it follows that there exists  := sup    in  and ‖‖ = sup ≥1 ‖  ‖.Again,  is said to have the Fatou property if this is the case for upwards directed systems.
An operator  :  →  between Banach lattices is said to be an order isometry if it is a linear isometry which is also an order isomorphism (i.e.,  is also one to one, onto, ‖‖  = ‖‖  for all  ∈  and ( ∧ ) =  ∧  for all ,  ∈ ).In this case  and  are said to be order isometric.

Integration with respect to Vector Measures on 𝛿-Rings.
We recall here the integration theory with respect to vector measures introduced by Lewis in the context of -algebras [16] and extended by Masani and Niemi and Stefansson in the context of -rings [17][18][19].We refer also to [2,20].
Given R a -ring of subsets of an abstract set Ω, that is, a ring of sets of Ω closed under countable intersections, we can consider R loc the associated -algebra to R defined by R loc = { ⊂ Ω :  ∩  ∈ R, for every  ∈ R}.Take the space M(R loc ) of measurable real functions on (Ω, R loc ) and denote by S(R loc ) and S(R) the space of simple functions with support in R loc and R, respectively.
Consider also ] : R →  a vector measure, that is, a set function from the -ring R to a real Banach space  for which ](∪    ) = ∑  ](  ) in the norm topology of  for all sequences (  ) of pairwise disjoint members of R such that ∪    ∈ R. Recall the definition of the semivariation of ] which is given by ‖]‖ : R loc → [0, ∞] with ‖]‖() = sup{| * ]|() :  * ∈   * } for all  ∈ R loc , where  * denotes the topological dual of ,   * denotes the unit ball in  * , and | * ]| is the usual variation of scalar measure  * ] : R → R. The semivariation of ] is monotone increasing, countably subadditive, finite on R, and satisfying for all  ∈ R loc , We will say that a set  ∈ R loc is ]-null if ‖]‖() = 0 and that a property holds ]-almost everywhere (]-a.e.) if it holds except on a ]-null set.Also, a vector measure ] : R →  with values in a Banach lattice  is positive if ]() ≥ 0 for all  ∈ R.
Taking into account the previous definitions, the integrability with respect to a vector measure is defined as follows.First, consider the space for all  * ∈  * (i.e., which are integrable with respect to | * ]| for all  * ∈  * ), where functions which are equal ]-a.e. are identified.Endow this space with the norm and the ]-a.e.pointwise order.Then  1  (]) is a Banach lattice containing S(R) in which convergence in norm of a sequence implies ]-a.e.convergence of some subsequence (see [18,Lemma 3.13]).Even more, the space  1  (]) is an ideal of measurable functions in the sense that, if || ≤ || ]-a.e. with  ∈ M(R loc ) and  ∈  1  (]), it follows that  ∈  1  (]).Now, take the closed ideal  1 (]) in  1 (]) of functions which are integrable with respect to ], that is, the functions in  1  (]) for which there exists a vector denoted by ∫  ] ∈ , such that for each  ∈ R loc .Then  1 (]) is a Banach lattice with the norm and the order inherited from  1  (]) in which S(R) is dense.The space  1 (]) is also an ideal of measurable functions, and the space  1 (]) coincides with  1  (]) whenever  does not contain any linear subspace isomorphic to  0 [16,Theorem 5.1].

Representation of Functionals on Spaces of Integrable Functions with respect to a Vector Measure on a 𝛿-Ring
Let R be a -ring and ] : R →  be a vector measure.Following [21, Section 2], we say that a countably additive measure  : R → [0, ∞] is a local control measure for ] if it satisfies the following conditions: (1) lim ⊂, () → 0 ‖]()‖  = 0, for every  ∈ R, The first condition is equivalent to ]() = 0 whenever  ∈ R with () = 0 (see [ In the proof of [21, Theorem 3.2], it is shown that there exists a maximal family {  } ∈Δ of nonnull sets members of R satisfying that   ∩   is ]-null for  ̸ = .Considering that, for each  ∈ Δ the class of sets R ∩   := { ∈ R :  ⊂   }, we have that R ∩   is a -ring as a collection of subsets of Ω and a -algebra as a collection of subsets of   .Taking ]  the restriction of ] to R ∩   , it is known that there always exists a finite local control measure   (called a Rybakov control measure and of type | * ]  |) for ]  .For this measure and for each  ∈ R, it is proved that   (  ∩ ) = 0 for all  ∈ Δ except on a countable set.Now, the set function  : R → [0, ∞] given by () = ∑ ∈Δ   ( ∩   ) is a local control measure for ].
We start our study looking for a suitable definition for the Köthe dual space of  1 (]).Not only the existence of a local control measure  for ] but also its particular construction will be the key of our work.Firstly, we establish a useful representation of functionals on spaces  1 (]).So, let R be a -ring and consider a (countably additive) vector measure ] : R → .Fix a decomposition {  } ∈Δ of the measure space as the one explained previously and take a countable family {   } ∈N of elements of {  } ∈Δ .In order to avoid misunderstandings, we advise that, throughout all the paper, the families {   } ∈N are supposed to be ]-a.e.disjoint; that is, all the indexes   are supposed to be different.Let  be a local control measure for ] with the properties of the ones constructed by Brooks and Dinculeanu.Since by the construction  is finite in all  ∈ R, we have that it is -finite in  := ⋃ ∈N    ∈ R loc .Lemma 1.Let {   } ∈N in {  } ∈Δ and  := ⋃ ∈N    as explained above.Then there is a finite measure   : R loc ∩  → R such that, for each  ∈ ( 1 (])) * , there is a measurable function  , ∈  0 (  ) such that for all  ∈  1 (]) so that supp{} ⊆ .
Proof.Let  be a fixed local control measure for ] constructed as explained previously.For each  ∈ N, consider the measure   : R ∩  → R given by   := |    .Note that all these measures are finite by the construction of .Consider the finite measure   : R loc ∩  → R given by   () := ∑ ∞ =1 (  (   ∩ )/2    (   )),  ∈ R loc ∩ .Define also the finite measure  , : R loc ∩  → R by  , () := ∑ ∞ =1 ((    ∩ )/2  ‖]‖(   )),  ∈ R loc ∩ .Note that this measure is well defined and absolutely continuous with respect to   : R loc ∩  → R. By the Radon-Nikodym theorem, there is an integrable function  0 , ∈  1 (  ) such that Thus, for each  ∈ S(R), we have that Take the function , and note that, for such an R-simple function , we have that Thus, for such a function  we obtain that Write  , := ℎ   0 , .A direct calculation using the order continuity of  1 (]) gives the result for this function  , .
Notice that the measure λ is well defined; although the measure  is defined just for elements of R, using classical arguments, it can be immediately seen that this formula extends it to the whole (R  ) loc .
Note also that R  = R ∩  and remark that (R  ) loc is computed with respect to the target space  that is, (R  ) loc := { ⊆  :  ∩  ∈ R  ,  ∈ R  }.In order to clarify the relation between the measurability with respect to the different -algebras involved, let us establish the following notation.If  is a subset of R loc , we write (R  ) loc  for the algebra associated with R  using as target space the set .No explicit reference is made when  = Ω.The relation between (R  ) loc  and R loc is now done by the following lemma.The proof is straightforward.
For a given function  ∈  1 (]), we can find a countable set {  } ∈N of different elements of Δ such that the support of  outside   = ⋃     is ]-a.e.null (and so ||(Ω \   ) = 0; see [20,Theorem 3.6]), and we can consider the -finite measure λ := λ  .Therefore, for a given function  ∈  1 (]), we find a set   , a -ring R   , and a -finite measure λ .
We describe now the adequate family of functions-like objects necessary to define the Köthe dual of an abstract space  1 (]).For  ∈ Δ, define the sets  0  () of classes of -a.e.equal R ∩   -measurable functions in R   .Consider the Cartesian product ∏ ∈Δ  0  ().For each countable set  of subsets    , consider   an element of ∏ ∈Δ  0  () that is a measurable function in each    and 0 out of .Take the vector space F 0 -with the obvious sum and product-of the families  := {  } of elements   ∈ ∏ ∈Δ  0  () that are compatible; that is, for each couple of countable families  and   , if there is an  ∈ Δ such that   belongs both to  and   , then the respective "-coordinates" (  )   and (   )   coincide |   -a.e.A natural norm-type function to consider for the elements of this space is given by countable union of    } ,  ∈ F 0 . ( It can be easily seen that this expression is a norm for the space F of all the elements of F 0 for which ‖ ⋅ ‖ F is finite.This construction allows to write the extension of the definition of Köthe dual for the setting of the abstract  1 (]) spaces as (F, ‖ ⋅ ‖ F ); that is, we define with the dual norm ‖‖ ( 1 (]))  := ‖‖ F . ( In what follows, we will consider a slightly different context in order to obtain a better representation of the Köthe dual space defined as in previous.Using the fixed decomposition given by {  } ∈Δ , define the -ring R 0 of the elements  = ⋃  =1   ,   ∈ R such that there is an   ∈ Δ with   ⊆    ,  ∈ N. Note that R 0 ⊆ R and so we have that R loc ⊆ (R 0 ) loc .We can then define the measure  : R 0 → R given by () := ∑  =1 (  ), where  ∈ R 0 is decomposed as mentioned earlier.The variation || of this measure is well defined and allows its extension to (R 0 ) loc .Note also that M(R loc ) ⊆ M(R 0 ) loc .
Proposition 3.For a vector measure ], consider a maximal R-decomposition {  } ∈Δ .Then and the norm for this space can be computed by Proof.It is a consequence of the following equivalences.Let  be a countable union of elements    .If  ∈ F and  ∈  1 (]) such that supp{} ⊆ , the function  ⋅    can be supposed to be 0 outside the set   , and so Write   for the measure    (⋅) = (⋅∩   ), where    ∈   , and note that Therefore, integrability of with respect to λ is equivalent to integrability of   with respect to ||, and the proof is complete.
Theorem 4. For a vector measure ], ( isometrically, where ( 1 (])) is defined from any maximal R- Proof.It is a consequence of Lemma 1, and Proposition 3. Let us show the proof in two steps.
Step 1.Let  ∈ ( 1 (])) * and  = ⋃ ∞ =1    .Then using Lemma 1 we find a function  , representing the Radon-Nikodym derivative of the set function (⋅|  )-that is a measure by the order continuity of  1 (])-in a way that for all  ∈  1 (]) such that supp{} ⊆ .Therefore, computing in each  the Radon-Nikodym derivative   of the measure   appearing in Lemma 1, with respect to λ , and writing whenever supp{} ⊆ , we obtain a family  of the space F 0 .Moreover, note that for such , we have that and so  ∈ F. In fact, it can be easily seen that the identification    is linear, continuous, and injective.
Step 2. Let us show now that this identification is also surjective.We use Proposition 3. Let  be a compatible family of elements of F 0 such that ‖‖ F < ∞.For each  ∈ as a consequence of the compatibility of the elements of F 0 that define the family .This allows to prove that  is well defined and linear; the finiteness of ‖‖ F gives also that sup ∈  1 (]) |()| ≤ ‖‖ F < ∞, and so  is an element of the dual space ( 1 (])) * .
Remark 5. Notice that ( 1 (])) is a Banach lattice with the order of ( 1 (])) * .It can be easily seen that it coincides with the natural order in F 0 .
Remark 6.Note that, for the case that the -ring R is a countable union of -algebras (i.e., the -finite case), a better representation for the Köthe dual is possible.Instead of defining it as the space of families of countable supported elements of a Cartesian product, a single function can play the role; that is, This is a trivial consequence of the definition of the families, since in the -finite case just a function defines a family.Taking into account the definition of the measures involved in the expression above, it is clear that this representation equals that is the well-known definition of Köthe dual for the -finite case.
We extend now the definition of the Köthe dual to abstract Banach lattices.Let  be an order continuous Banach lattice.Then it is possible to define its Köthe dual by means of an integral representation (using vector measures over -rings) which can be described as follows.First, by [12, Proposition 1.a.9],  can be decomposed into an unconditionally direct sum of a family of mutually disjoints ideals {  } ∈Δ , each   having a weak unit.Each   is now an order continuous Banach lattice with a weak unit and so, from [22,Theorem 8], there exist a -algebra Σ  of parts of an abstract set Ω  and a positive vector measure ]  : Σ  →   such that the integration operator  ]  :  1 (]  ) →   is an order isometry.Consider the set Ω = ⋃ ∈Δ {} × Ω  and the -ring R of subsets of Ω given by the sets ⋃ ∈Δ {} ×   satisfying that   ∈ Σ  for all  ∈ Δ and there exists a finite set  ⊂ Δ such that   is ]  -null for all  ∈ Δ \ .Then, take the vector measure ] : R →  defined by and compute the integration operator  ] :  1 (]) → .This operator is an order isometry (see [1,Theorem 4] and [3, pp. 22-23]).Let (Ω, (R 0 ) loc , ||) be the measure space associated with ].The definition of the Köthe dual of  is done using this integral representation adequately: Corollary 7. Let  be an order continuous Banach lattice.Let ] be the vector measure which appears in the representation explained previously.Then  * =   .
Let us prove now a related result regarding the Köthe dual of  1 (]).The representation of the Köthe dual of  1 (]) given in Proposition 3 suggests the following definition of Köthe dual space for any Banach lattice  of ||-a.e.classes of (Ω, (R 0 ) loc )-measurable functions, in particular, for  1  (]).We define the Köthe dual of  as their Köthe duals coincide and are equal to ℓ 1 (Γ), providing in this way a description of the Köthe dual of the relevant space ℓ ∞ 0 (Γ).
(2) Although the construction of (1) regards discrete vector measures, the results in this paper can also be applied for general Banach lattices of functions involving measures that are not discrete.Let us write the Köthe dual of the space appearing in Remark 22 of [4].Consider a family of disjoint probability spaces (Ω  , Σ  ,   ) ∈Γ , where Γ is an uncountable set of indexes.The -ring R is defined by finite unions  = ⋃  =1    ,    ∈ Σ   .We consider the vector measure  : R →  0 (Γ) given by () = ∑  =1    (   ) {  } .As in example 2.2 of [20], it can be easily shown that the space  1 () is the direct sum ⨁  0 (Γ)  1 (  ).Consequently, the support of an element of this space is contained in a countable union of sets Ω  ,  ∈ Γ.However,  1  () = ⨁ ℓ ∞ (Γ)  1 (  ) and the functions of this space can be even strictly positive in all points of ⋃ ∈Γ Ω  .This is suggested by the symbols ⨁  0 (Γ) and ⨁ ℓ ∞ (Γ) that we use: in the first case, countability of the support of each element is assumed but not in the second.Thus, we can describe an element  ∈  1 () as a sequence (   ) ∞ =1 , where each    ∈  1 (   ).Let us consider the variation || of the local control measure  for  that is defined for each element  = ⋃  =1    ∈ R as () = ∑  =1    (   ).The Köthe dual of  1 () is then given by ( 1 ()) where each   of an element  of F can be identified with an element of the direct sum ⨁ ℓ 1 (Γ)  ∞ (  ) in the usual manner.Note that, in this case, R and R 0 coincide.After Theorem 4, we know that this provides a description of the dual space ( 1 ()) * .By Corollary 8, this space coincides also with ( 1  ())  .
, for each countable union  of sets    ,   ∈ Δ, such that   ⊆ ,