On finite elements in vector lattices and Banach lattices

Transcription

1 Math. Nachr. 279, No. 5 6, (2006) / DOI /mana On finite elements in vector lattices and Banach lattices Z. L. Chen 1 and M. R. Weber 2 1 Department of Mathematics, Southwest Jiaotong University, Chengdu , P. R. China 2 Fachrichtung Mathematik, Technische Universität Dresden, Dresden, Germany Received 18 February 2004, revised 7 June 2004, accepted 17 June 2004 Published online 9 March 2006 Key words Vector lattice, Banach lattice, finite element, order continuous norm MSC (2000) 46B42, 47B07, 47B65 In Archimedean vector lattices we show that each element of the band generated by a finite element is also finite. In vector lattices with the (PPP) and in Banach lattices we obtain some characterizations of finite elements by using the generalized order units for principal bands. In the case of Banach lattices with order continuous norm the ideal of all finite elements coincides with the linear span of all atoms. Some other related results and applications are included. 1 Introduction In vector lattices of continuous functions on a locally compact Hausdorff space the finite functions, i.e., continuous functions with compact support, play a natural and important role. The notion of a finite element (see Definition 1.1) in an Archimedean vector lattice as some abstract analogon of functions with compact support has been introduced and thoroughly studied in [4]. An Archimedean vector lattice possessing a sufficient number of finite elements allows (under some additional conditions) a representation as a vector lattice of (everywhere finite-valued) continuous functions on a locally compact σ-compact space such that all finite elements are represented as finite functions. For details of representations and for a topological characterization of finite and totally finite elements (see Definition 1.2) we refer to [4] [7], [9], [12]. Further on E (or, if the cone E + has to be explicitly mentioned, (E,E + )) always denotes an Archimedean vector lattice. We use the Archimedean property as follows: if rx y for all r R + then x 0. In Section 3 the vector lattice E will be equipped with a lattice norm such that (E,E +, ) is a Banach lattice. Definition 1.1 An element ϕ E is called finite, if there is an element z E satisfying the following condition: for any element x E there exists a number C x > 0 such that the inequality x n ϕ C x z (1.1) holds for all n N. The element z is called an E-majorant or briefly a majorant of the finite element ϕ. The set of all finite elements of a vector lattice E is denoted by Φ 1 (E). Obviously, Φ 1 (E) is an ideal, i.e., a solid (sometimes also called normal) linear subspace of E. The special class of finite elements characterized by possessing at least one E-majorant which itself is a finite element turns out to be in general different from Φ 1 (E). Definition 1.2 A finite element ϕ E is called totally finite if it has an E-majorant z belonging to Φ 1 (E). The set of all totally finite elements of a vector lattice E is also an ideal which is denoted by Φ 2 (E). It is clear that Φ 1 (E) =E implies Φ 2 (E) =Φ 1 (E). Obviously there hold the inclusions {0} Φ 2 (E) Φ 1 (E) E zlchen@home.swjtu.edu.cn, Phone: +86 (0) , Fax: +86 (0) Corresponding author: weber@math.tu-dresden.de, Phone: +49 (0) , Fax: +49 (0)

2 496 Chen and Weber: Finite elements in vector lattices which might be proper (see [12]). In [5] the relation Φ 1 (Φ 2 (E)) = Φ 2 (E) was shown, that means any element φ Φ 2 (E) possesses an E-majorant which itself is a totally finite element. It is now of interest to study and characterize finite and totally finite elements in some vector lattices. Although some general facts on finite elements in Dedekind complete vector lattices have been used to prove some results for the (Dedekind complete) vector lattice M(T ) of Radon measures 1 on a locally compact Hausdorff space T in [10] (see also [11]) there is no systematic study of this type of elements in Dedekind complete vector lattices, or as the formula (1.1) might indicate, in vector lattices satisfying the principal projection property (see below) or even in more general vector lattices. Recall some definitions, notations and elementary facts in an Archimedean vector lattice (E,E + ) which will be used further on. For details see [1] and [8]. First of all a set B E is called a band if it is an order closed ideal, the limit (in E) of any order convergent net of the ideal B belongs to B. Two elements x, y E are called disjoint, written as x y,if x y =0. For any nonempty subset B E denote by B d the set {x E : x y for any y B}. ThesetB dd is known as the band generated by B, the smallest band that contains B. IfB consists of one element x, the band generated by {x} is called principle band. AbandB in E is a projection band if E = B B d. In this case any element x E has a unique representation x = x 1 + x 2,wherex 1 B and x 2 B d.themapp B : E E defined by P B (x) =x 1 for any x E = B B d is a positive projection. In a Dedekind complete vector lattice any band is a projection band. If {u} dd is a projection band then P {u} dd is denoted by P u. In this case for each element x 0 there exists the element sup{x n u } and, P u (x) (for x 0) is calculated by the formula P u (x) = sup{x n u }. (1.2) A vector lattice E is said to have the principal projection property (PPP), if {x} dd is a projection band for each x E. Any σ-dedekind complete vector lattice has the (PPP). An element u E + is an order unit, if for each x E there is a λ R with λu x λu (or equivalently, x λu). An element e E + is a weak order unit,ifx E and x e imply x =0. An element 0 <a E is called an atom of E, if whenever 0 b a one has b = λa. A Banach lattice is said to be atomic if for each x>0 there is an atom a such that 0 <a x. 2 Finite elements in vector lattices We start with a basic result Proposition 2.1 If a vector lattice E possesses an order unit, then each element x E is finite, Φ 1 (E) = Φ 2 (E) =E. Proof. Ifu E + is an order unit, then for each x E there is a positive number C x such that x C x u. For any v E one has x n v x C x u which, by Definition 1.1, shows that v is a finite element. Example 2.2 If E is one of the vector lattices c, l, L (µ) and C(K) (where K is a compact Hausdorff space) then Φ 1 (E) =Φ 2 (E) =E. Proposition 2.3 Let E be vector lattice. Then each atom of E is a totally finite element with itself as a majorant. 1 If K(T ), the vector space of all continuous functions on T with compact support, is equipped with the topology of the inductive limit, then M(T ) coincides with the linear functionals continuous with respect to this topology, i.e., (K(T )) = M(T ).

3 Math. Nachr. 279, No. 5 6 (2006) 497 Proof. Leta be an atom of E. For each x E + define λ a (x) = sup{r R + : ra x}. (2.1) Clearly λ a (x) < for all x E + as E is Archimedean. Now for x E and every n one has x na na, 1 n ( x na) a and, by taking into consideration that a is an atom, x na = r na x for some r n,which forces that x na λ a ( x )a. Therefore a is finite. Let X be a given vector sublattice of a vector lattice E. Anelementz E + is called a generalized order unit for X, if for each x X there is a real number C x > 0 with x C x z. Note that X then belongs to the ideal generated (in E) by z and that z is not required to belong to X + = X E +. Theorem 2.4 Let E be a vector lattice. If φ E is a finite element then {φ} dd has a generalized order unit and {φ} dd Φ 1 (E). Proof. Ifφ E is finite then there exists z E + such that for each x E there is a real number C x > 0 with x (n φ ) C x z for all n N. It follows from [1, Theorem 3.4] that x = sup{ x n φ } C x z for all x {φ} dd, which implies that the element z is a generalized order unit of {φ} dd. Now for each v {φ} dd and arbitrary x E, since x n v {φ} dd, again by the same Theorem 3.4 one has x n v = sup{ x n v m φ : for all m N} = sup{( x m φ ) n v : for all m N} (C x z) (n v ) C x z for all n. So,v is finite and {φ} dd Φ 1 (E). The next result shows that the converse of Proposition 2.1 is true if the vector lattice E has a weak order unit. Corollary 2.5 Let E be a vector lattice with a weak order unit. Then Φ 1 (E) =Φ 2 (E) =E if and only if E has an order unit. Proof. Indeed,if e is a weak order unit in E =Φ 1 (E) then by the theorem {e} dd has a generalized order unit which due to {e} dd = E is obviously an order unit. Now Proposition 2.1 completes the proof. Note that we have established that any generalized order unit of the band generated by the weak order unit serves as an order unit in E. However, a weak order unit of a vector lattice E fails to be an order unit in general, even if Φ 1 (E) =Φ 2 (E) = E and E has an order unit. For example if E = c and u = ( 1, 1 2,..., 1 n,...) then u is a weak order unit of E but it is not an order unit. The next result is a characterization of finite elements in vector lattices with (PPP). Note that the equivalence 1) 3) in case of a Dedekind complete vector lattice was proved in [11, Theorem 1]. Theorem 2.6 Let E be a vector lattice with (PPP) and φ E. Then the following statements are equivalent: 1) φ is a finite element of E. 2) {φ} dd has a generalized order unit z E +. 3) {φ} dd has an order unit z 0 {φ} dd.

4 498 Chen and Weber: Finite elements in vector lattices Proof. 1) 2) is precisely Theorem ) 3): Ifz E + is a generalized order unit of {φ} dd then for each x {φ} dd, there is a real positive number C x such that x C x z.letp φ be the band projection from E onto {φ} dd. It follows that x = P φ x P φ (C x z) = C x P φ z = C x z 0, where z 0 = P φ z {φ} dd, which implies that z 0 is an order unit of {φ} dd. 3) 1): Ifz 0 is an order unit of {φ} dd then for arbitrary x E, sincep φ x {φ} dd, there is a positive number C x such that P φ x C x z 0,sothat x n φ sup{ x n φ } = P φ x C x z 0 for all n N. This implies that φ is a finite element of E. Remarks 2.7 (1) It has been proved a little more. Namely, if φ is a finite element and z an arbitrary one of its generalized order units then P φ z is an order unit in {φ} dd. If, in addition, {φ} dd Φ 1 (E) then {φ} dd Φ 2 (E). (2) The proof also shows that the finiteness of an element in an arbitrary Archimedean vector lattice can be detected by the properties of its principle band (as was mentioned by A. I. Veksler): Let E be an arbitrary Archimedean vector lattice and let the element φ E be such that {φ} dd is a projection band. Then φ is a finite element (with the majorant z) if and only if {φ} dd contains an order unit (namely, P φ z). In particular, if E is a σ-dedekind complete vector lattice then Φ 1 (E) =E if and only if each principal band possesses an order unit. Combining Theorems 2.4 and 2.6 we have the following Theorem 2.8 Let E be a vector lattice with (PPP).ThenΦ 1 (E) =Φ 2 (E) and Φ 1 (E) also has the (PPP). Proof. Letφ Φ 1 (E) and let z be its majorant. Then P φ z is also a majorant for φ. Since by Theorem 2.4 {φ} dd Φ 1 (E), wegetφ 1 (E) =Φ 2 (E). Ifv Φ 1 (E) then {v} dd is a projection band in E which is a subset of Φ 1 (E). This shows that the ideal Φ 1 (E) has (PPP). 3 Finite elements in Banach lattices At present we do not know whether or not the condition {φ} dd has a generalized order unit is sufficient for φ to be a finite element. In order to answer this and some other natural questions, we restrict the investigation to a smaller class of vector lattices. In this part we equip the Archimedean vector lattices with the additional structure of a Banach space. So (E,E +, ) denotes now a Banach lattice. The stated question (the converse of Theorem 2.4) is answered affirmatively if E is a Banach lattice. The norm in a Banach lattice is said to be order continuous if x γ 0 implies x γ 0. Banach lattices with order continuous norm are characterized, e.g., in [8, Theorem 2.4.2], in particular, such Banach lattices are Dedekind complete. Theorem 3.1 Let E be a Banach lattice and φ E. Thenφ is finite if and only if {φ} dd has a generalized order unit. P r o o f. We only need to show that, if {φ} dd has a generalized order unit then the element φ is finite. In fact, let z E + be a generalized order unit of {φ} dd. Define a norm by x z = inf{λ >0: x λz} for each x {φ} dd. Then by [1, Theorem 12.20] ( {φ} dd, z ) is an AM-space, where x x z z holds. By [8, Proposition 1.2.3] the band {φ} dd is closed in E so that ( {φ} dd, ) also is a Banach space. Now the open mapping theorem implies that the norms and z are equivalent on {φ} dd, thus in particular, there is a constant C > 0 such that x z C x for all x {φ} dd.since x x z z for each x {φ} dd,wehave x z C, i.e., x Cz, for each x {φ} dd with x 1. Now for arbitrary x E it follows from 0 x n φ x (and hence x n φ x ) that x n φ x Cz for all n, which means that φ is finite.

5 Math. Nachr. 279, No. 5 6 (2006) 499 Remarks 3.2 (1) In the proof we have used the following fact. If in {φ} dd the two norms and z are equivalent and a set A {φ} dd is -bounded then A is order bounded. (2) The principal band generated by a finite element may fail to possess an order unit as the following example shows. Let E = C[0, 1] and H c = {x E : x(t) =0, for all t [0,c]} for each c (0, 1),then (i) Φ 1 (E) =E. (ii) H c is a principal band for each c (0, 1), moreover, H c = {φ} dd for any φ H c satisfying φ(t) 0 for t (c, 1]. (iii) H c does not possess any order unit but each function z E with z(t) > 0 for t (c δ, 1] is a generalized order unit, where δ is some positive number. For a Banach lattice E we use Γ E to denote the set of all atoms of E with norm 1. From Proposition 2.3 there follows that Γ E Φ 1 (E). It is not difficult to verify that Γ E consists of pairwise disjoint elements, hence Γ E is linear independent. If E is atomic then x = sup{λ a (x)a : a Γ E } for each x E +, where λ a (x) is defined by (2.1). Indeed, clearly x λ a (x)a for any a Γ E.Ifz λ a (x)a then z x λ a (x)a for all a Γ E. If one would have x z x>0then there exist b Γ E and r>0such that rb x z x (due to E be atomic). Together with z x λ b (x)b there follows (r + λ b (x))b x z x + z x = x. The definition of λ b (x) implies now r + λ b (x) λ b (x), a contradiction. This means x = z x z. If, in addition, E has an order continuous norm then λ a (x) =0for all but countable many a Γ E and x = a Γ E λ a (x)a for each x E +, hence E = span(γ E ), where the series is norm convergent. Indeed, for any n N the set Γ n = { a Γ E : λ a (x) n} 1 is finite due to the order continuity of the norm. So {a ΓE : λ a (x) > 0} = n=1 Γ n is at most countable, say {a 1,...,a n,...}. Then λ a1 (x)a λ an (x)a n x. Again by order continuity of the norm we get x = n=1 λ a n (x)a n, for more details see [2]. Theorem 3.3 Let the norm of the Banach lattice E be order continuous. Then 1) Φ 1 (E) =Φ 2 (E) =span(γ E ). 2) Φ 1 (E) is closed in E if and only if Γ E is a finite set, particularly, Φ 1 (E) =E if and only if E is finite dimensional. Proof. 1) Φ 1 (E) =Φ 2 (E) is clear from Theorem 2.8 since E is Dedekind complete. As it was already mentioned each a Γ E is a finite element of E, sospan(γ E ) Φ 1 (E) since Φ 1 (E) is a sublattice 2 (and an ideal) of E. Forφ E is a finite element, Theorem 2.6 implies that the principal band {φ} dd has an order unit. By [8, Proposition and Corollary ] and Kakutani s theorem {φ} dd is lattice isomorphic to an AMspace C(K) for some compact Hausdorff space K. It follows that the lattice operations on {φ} dd are weakly sequentially continuous as the lattice operations in each AM-space are (see [8, Proposition ]), and the closed unit ball of {φ} dd is order bounded. Observe that {φ} dd also is a Banach lattice with an order continuous norm, so [3, Corollary 2.3] implies that {φ} dd is atomic. From the above we have {φ} dd = span(h), where H =Γ E {φ} dd. The closed unit ball of {φ} dd, as a subset of an order interval, is compact (see [13, Theorem 5]), therefore {φ} dd is finite dimensional and hence H is a finite set, {φ} dd =span(h) span(γ E ),which means that Φ 1 (E) span(γ E ). 2) Based on the first part of the theorem it is clear that Φ 1 (E) is not closed in E if Γ E is infinite. Immediately from the theorem we obtain some information on the set of all finite elements in the following Banach lattices: Corollary 3.4 1) If E = c 0 or l p with 1 p< then Φ 1 (E) =Φ 2 (E) =span{e k : k =1, 2, 3,...}, where e k E with k s entry equal to 1 and all others are 0. 2) If E = L p (a, b) with 1 p< then Φ 1 (E) ={0}. In [12, case d] the vector lattice of all finite continuous functions on (, + ) was mentioned as an example of a vector lattice E possessing the property Φ 1 (E) =Φ 2 (E) =E. The next result indicates an abstract class of vectorlattices E with this property. 2 If ΓE = then we define span(γ E )={0}.

6 500 Chen and Weber: Finite elements in vector lattices Theorem 3.5 For a Banach lattice E the following statements are equivalent: 1) Φ 1 (E) =Φ 2 (E) =E. 2) E is lattice isomorphic to an AM-space and each principal band has a generalized order unit. Proof. 2) 1) follows straightforward from Theorem ) 2). IfΦ 1 (E) =E then due to Theorem 2.4 it suffices to show that E is lattice isomorphic to an AMspace. According to [8, Theorem ] it suffices to show that {x n } is order bounded in E whenever x n E + with x n 0. Putu = n=1 1 2 x n n,thenx n {u} dd for each n. Sinceu is finite it follows from Theorem 3.1 and Remark 3.2 (1) that the sequence {x n } is order bounded in E as {x n } is norm bounded in {u} dd. The following example shows that E mayfailtohaveanorderunitevenife is a Dedekind complete AMspace and Φ 1 (E) =Φ 2 (E) =E. Example 3.6 Let J be an uncountable index set and E the space of all bounded real functions f on J such that f(j) =0for all but countable many j J. Under the pointwise defined algebraic operations, the pointwise order and equipped with the supremum norm, it is easy to verify that E is a Dedekind complete AM-space without order unit. But each principal band obviously has an order unit, so it follows that Φ 1 (E) =Φ 2 (E) =E. For a vector lattice E denote by L r (E) the vector space of all regular operators on E, where a linear operator T( belongs to L r (E) iff T = T 1 T 2 for T i 0, i=1, 2. An operator T on E is said to be order bounded T L b (E) ) if it maps any order bounded set into an order bounded set. In general one has L r (E) L b (E). For Dedekind complete vector lattices both classes of operators coincide. An order bounded operator T on E is called orthomorphism (T Orth(E)) if T leaves each band of E invariant, i.e., whenever T (B) B holds for each band B E. As an application of Theorem 2.6 we have now Theorem 3.7 Let E be a Dedekind complete Banach lattice. Then Orth(E) Φ 1 (L r (E)) = Φ 2 (L r (E)). P r o o f. By the Riesz Kantorovich Theorem the vector lattice L r (E) is Dedekind complete 3 and therefore, due to Theorem 2.8, Φ 1 (L r (E)) = Φ 2 (L r (E)). According to [1, Theorem 8.11] the collection Orth(E) is the band generated by identity operator I in L r (E) and Orth(E) = {I} dd = {T L r (E) : λi T λi}. By [1, Theorem 15.5] the vector lattice Orth(E) equipped with the norm T I = inf{λ >0: T λi} turns out to be an AM-space with unit. Now the result follows from the Theorems 2.6 and 2.4 above. Acknowledgements The first named author was supported by China Scholarship Council during a stay for a half year at the Technical University of Dresden. References [1] C. D. Aliprantis and O. Burkinshaw, Positive Operators (Academic Press, Inc., London, 1985). [2] Z. L. Chen, Vector lattices of regular operators on Banach lattices, Acta Math. 43 (2), (2000) (in Chinese). [3] Z. L. Chen and A. W. Wickstead, Relative weak compactness of solid hulls in Banach lattices, Indag. Math. (N.S.) 9, (1998). [4] B. M. Makarow and M. Weber, Über die Realisierung von Vektorverbänden I, Math. Nachr. 60, (1974). [5] B. M. Makarow and M. Weber, Einige Untersuchungen des Räumes der maximalen Ideale eines Vektorverbandes mit Hilfe finiter Elemente I, Math. Nachr. 79, (1977). [6] B. M. Makarow and M. Weber, Einige Untersuchungen des Räumes der maximalen Ideale eines Vektorverbandes mit Hilfe finiter Elemente II, Math. Nachr. 80, (1977). [7] B. M. Makarow and M. Weber, Über die Realisierung von Vektorverbänden III, Math. Nachr. 86, 7 14 (1978). [8] P. Meyer Nieberg, Banach Lattices (Springer-Verlag, Berlin Heidelberg New York, 1991). 3 The order in L r (E) is introduced by means of all positive operators on E.

7 Math. Nachr. 279, No. 5 6 (2006) 501 [9] M. Weber, Über die Realisierung von Vektorverbänden II, Math. Nachr. 65, (1975). [10] M. Weber, Finite Elemente im Raum der Radonmaße, Optimizacia 47, (1990) (in Russian). [11] M. Weber, Finite Elemente im Vektorverband der Radonmaße, Wiss. Z. Tech. Univ. Dresden 41 (5), (1992). [12] M. R. Weber, On finite and totally finite elements in vector lattices, Anal. Math. 21, (1995). [13] A. W. Wickstead, Compact subsets of partially ordered Banach spaces, Math. Ann. 212, (1975).