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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 782792, 8 pages
The Köthe Dual of an Abstract Banach Lattice
1Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva S/N, 18071 Granada, Spain
2Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera S/N, 46022 Valencia, Spain
Received 28 May 2013; Accepted 6 June 2013
Academic Editor: Miguel Martin
Copyright © 2013 E. Jiménez Fernández et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 of integrable functions and 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 -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 (see [1, Theorems 4 and 8] and ; 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 [4–6]).
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 for an uncountable set of indexes 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 -space (an -space) of a vector measure are nowadays known (see [7–10]). 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 the mathematical literature regarding the notion of the space of the integrals as elements of the dual space.
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 there exists an increasing sequence so that in . If this property holds by means of upwards directed systems, is said to be order dense in . An element such that implies 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 it follows that . 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 in with , it follows that there exists in and . 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.
2.2. Integration with respect to Vector Measures on -Rings
We recall here the integration theory with respect to vector measures introduced by Lewis in the context of -algebras  and extended by Masani and Niemi and Stefansson in the context of -rings [17–19]. We refer also to [2, 20].
Given a -ring of subsets of an abstract set , that is, a ring of sets of closed under countable intersections, we can consider the associated -algebra to defined by , for every . Take the space of measurable real functions on and denote by and the space of simple functions with support in and , respectively.
Consider also a vector measure, that is, a set function from the -ring to a real Banach space for which in the norm topology of for all sequences of pairwise disjoint members of such that . Recall the definition of the semivariation of which is given by with for all , where denotes the topological dual of , denotes the unit ball in , and is the usual variation of scalar measure . The semivariation of is monotone increasing, countably subadditive, finite on , and satisfying for all , We will say that a set is -null if and that a property holds -almost everywhere (-a.e.) if it holds except on a -null set. Also, a vector measure with values in a Banach lattice is positive if for all .
Taking into account the previous definitions, the integrability with respect to a vector measure is defined as follows. First, consider the space of functions in such that 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 is a Banach lattice containing in which convergence in norm of a sequence implies -a.e. convergence of some subsequence (see [18, Lemma 3.13]). Even more, the space is an ideal of measurable functions in the sense that, if -a.e. with and , it follows that .
Now, take the closed ideal in of functions which are integrable with respect to , that is, the functions in for which there exists a vector denoted by , such that for each . Then is a Banach lattice with the norm and the order inherited from in which is dense. The space is also an ideal of measurable functions, and the space coincides with whenever does not contain any linear subspace isomorphic to [16, Theorem 5.1].
Regarding the lattice structure of these spaces, recall that has the -Fatou property, is order continuous, , and is always order dense in .
Finally, recall the integration operator given by which is linear and continuous with .
3. Representation of Functionals on Spaces of Integrable Functions with respect to a Vector Measure on a -Ring
Let be a -ring and be a vector measure. Following [21, Section 2], we say that a countably additive measure is a local control measure for if it satisfies the following conditions: (1), for every , (2)every -null set in is -null. The first condition is equivalent to whenever with (see [18, Proposition 3.6]), and this happens if and only if every -null set in is -null, due to the properties of the semivariation. Consequently, the countably additive measure is a local control measure for if and only if has the same null sets as . From [21, Theorem 3.2], there always exists such a measure, so and are Banach functions spaces related to . Here, by a Banach function space related to , we mean a Banach space satisfying that, if with and then and , where denotes the vector lattice of all measurable real functions on (functions which are equal -a.e. are identified) endowed with the -a.e. pointwise order.
In the proof of [21, Theorem 3.2], it is shown that there exists a maximal family of nonnull sets members of satisfying that is -null for . Considering that, for each the class of sets , we have that is a -ring as a collection of subsets of and a -algebra as a collection of subsets of . Taking the restriction of to , 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 , it is proved that for all except on a countable set. Now, the set function given by is a local control measure for .
We start our study looking for a suitable definition for the Köthe dual space of . 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 . So, let be a -ring and consider a (countably additive) vector measure . Fix a decomposition of the measure space as the one explained previously and take a countable family of elements of . In order to avoid misunderstandings, we advise that, throughout all the paper, the families 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 , we have that it is -finite in .
Lemma 1. Let in and as explained above. Then there is a finite measure such that, for each , there is a measurable function such that for all so that .
Proof. Let be a fixed local control measure for constructed as explained previously. For each , consider the measure given by . Note that all these measures are finite by the construction of . Consider the finite measure given by , .
Define also the finite measure by , . Note that this measure is well defined and absolutely continuous with respect to . By the Radon-Nikodym theorem, there is an integrable function such that Thus, for each , we have that Take the function , and note that, for such an -simple function , we have that Thus, for such a function we obtain that Write . A direct calculation using the order continuity of gives the result for this function .
(1) Let us consider now the following construction in order to define the Köthe dual of . Fix again a maximal -decomposition and a local control measure for . For every countable family in , we can define the following: the set , a -ring of subsets of by , and a countably additive positive measure given by , .
Notice that the measure is well defined; although the measure is defined just for elements of , using classical arguments, it can be immediately seen that this formula extends it to the whole .
Note also that and remark that is computed with respect to the target space that is, . 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 , we write for the -algebra associated with using as target space the set . No explicit reference is made when . The relation between and is now done by the following lemma. The proof is straightforward.
Lemma 2. Let . Then . In particular, the measurable functions in when restricted to are measurable with respect to , and when the functions are restricted to coincides with .
For a given function , we can find a countable set of different elements of such that the support of outside is -a.e. null (and so ; see [20, Theorem 3.6]), and we can consider the -finite measure . Therefore, for a given function , we find a set , a -ring , and a -finite measure .
We describe now the adequate family of functions-like objects necessary to define the Köthe dual of an abstract space . For , define the sets of classes of -a.e. equal -measurable functions in . Consider the Cartesian product . For each countable set of subsets , consider an element of that is a measurable function in each and out of . Take the vector space —with the obvious sum and product—of the families of elements that are compatible; that is, for each couple of countable families and , if there is an 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 It can be easily seen that this expression is a norm for the space of all the elements of for which is finite. This construction allows to write the extension of the definition of Köthe dual for the setting of the abstract spaces as ; that is, we define with the dual norm .
(2) 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 , define the -ring of the elements , such that there is an with , . Note that and so we have that . We can then define the measure given by , where is decomposed as mentioned earlier. The variation of this measure is well defined and allows its extension to . Note also that .
Proposition 3. For a vector measure , consider a maximal -decomposition . 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 and such that , the function can be supposed to be outside the set , and so . Then if and only if Write for the measure , where , and note that Therefore, integrability of with respect to is equivalent to integrability of with respect to , and the proof is complete.
Theorem 4. For a vector measure , isometrically, where is defined from any maximal -decomposition .
Proof. It is a consequence of Lemma 1, and Proposition 3. Let us show the proof in two steps.
Step 1. Let and . 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 —in a way that for all such that . Therefore, computing in each the Radon-Nikodym derivative of the measure appearing in Lemma 1, with respect to , and writing whenever , we obtain a family of the space . Moreover, note that for such , we have that and so . 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 such that . For each , define and notice that, for each countable union of sets , , such that , as a consequence of the compatibility of the elements of that define the family . This allows to prove that is well defined and linear; the finiteness of gives also that , and so is an element of the dual space .
Thus, we have proved that isometrically.
Remark 5. Notice that is a Banach lattice with the order of . It can be easily seen that it coincides with the natural order in .
Remark 6. Note that, for the case that the -ring 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 ], can be decomposed into an unconditionally direct sum of a family of mutually disjoints ideals , 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 of parts of an abstract set and a positive vector measure such that the integration operator is an order isometry. Consider the set and the -ring of subsets of given by the sets satisfying that for all and there exists a finite set such that is -null for all . Then, take the vector measure defined by and compute the integration operator . This operator is an order isometry (see [1, Theorem 4] and [3, pp. 22-23]). Let 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 . The representation of the Köthe dual of given in Proposition 3 suggests the following definition of Köthe dual space for any Banach lattice of -a.e. classes of -measurable functions, in particular, for . We define the Köthe dual of as endowed with the norm Let us show that, as in the finite measure case—that is, in the case of a vector measure on a -algebra—, the Köthe duals of and coincide. For a Banach function space on a -finite measure space, this is the well-known fact that —where is the order continuous part of —, whenever is super order dense in .
Corollary 8. Let be a vector measure on a -ring. Then one has that .
Proof. The inclusion is obvious. If , for each countable family , we have that for each .
Consider the restriction of to functions with support in . For every function with support in , we have that there is an increasing sequence in converging a.e. to (see the proof of [2, Theorem 4.8]). Notice also that, for all ,
If , we have that the restriction can be approximated -a.e. using an increasing sequence of simple functions with support in as previously mentioned: increases a.e. to where the sequence is clearly in . The monotone convergence theorem when applied to the sequence of gives that . Theorem 4 completes the proof.
Corollary 9. Let be a vector measure on a -ring and consider an ideal of containing . Then .
Let be a Banach lattice with the Fatou property having its -order continuous part as an order dense subset. As explained for the order continuous case, it is again possible to define its Köthe dual using a suitable representation for by means of a space of integrable functions. Let us describe briefly the elements we need. Consider the vector measure associated with as in Corollary 7, then is an order isometry. Extend now to the space as follows. For every , choose such that . For each and finite, take and consider in . Finally, define and extend to the general case. The extension defined in this way is again an order isometry between and (see [1, Theorem 8]). Consequently, the definition of the Köthe dual of can be done using this representation; that is, where is the measure space associated with .
Corollary 10. Let be a Banach lattice with the Fatou property having its -order continuous part as an order dense subset. Let be the vector measure which appears in the representation explained previously. Then .
4. Some Examples and Applications
(1) Consider and the space for a noncountable index set . It is an order continuous Banach lattice. Consider the -ring defined by the finite subsets of and the vector measure . Then the space can be directly computed and coincides with isometrically and in order (see example 2.2 in  and example 4.1 and the comments before Theorem 5.1. in ). Fixing the maximal decomposition of given by the singletons , and after Proposition 3, we get that its Köthe dual space is given by since is in this case , taking into account that the variation of the local control measure in this case can be taken to be the counting measure and also that obviously . Since the maximal decomposition that we are considering is given by all the singletons, we can identify each element with an element of , and so can be written as This characterization and Theorem 4 give that the dual space coincides with this Köthe dual space. This description has been frequently used in examples and computations of general Banach space theory. The reader can find these arguments for instance in  (see Remark 2.1), where the computation of the Köthe dual is based on the fact that the related measure space is discrete.
The same construction can be done for the case when the vector measure defined previously takes its values in . Then it is known that and . Thus the (Banach) lattice of the elements of with countable support satisfies . The constructive arguments given previously for instead of together with Corollaries 8 and 9 give that all the spaces appearing in the inclusion satisfy that their Köthe duals coincide and are equal to , providing in this way a description of the Köthe dual of the relevant space .
(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 . Consider a family of disjoint probability spaces , where is an uncountable set of indexes. The -ring is defined by finite unions , . We consider the vector measure given by . As in example 2.2 of , it can be easily shown that the space is the direct sum . Consequently, the support of an element of this space is contained in a countable union of sets , . However, and the functions of this space can be even strictly positive in all points of . This is suggested by the symbols and 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 as a sequence , where each . Let us consider the variation of the local control measure for that is defined for each element as . The Köthe dual of is then given by that can also be written as where each of an element of can be identified with an element of the direct sum in the usual manner. Note that, in this case, and coincide. After Theorem 4, we know that this provides a description of the dual space . By Corollary 8, this space coincides also with .
M. A. Juan was supported by Ministerio de Economía y Competitividad (Spain) (Project MTM2011-23164). E. Jiménez Fernández was supported by Junta de Andalucía and FEDER Grant (P09-FQM-4911) (Spain) and by Ministerio de Economía y Competitividad (Spain) (Project MTM2012-36740-C02-02). E. A. Sánchez-Pérez was supported by Ministerio de Economía y Competitividad (Spain) (Project MTM2012-36740-C02-02).
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