Finite Elements in Vector Lattices

Martin R. Weber (Dresden)

Abstract. In vector lattices of continuous functions on a locally compact Hausdorff space a natural andimportant role play the finite functions, i.e. continuous functions with a compact support.Finite and totally finite elements are the abstract notion ofsuch functions in arbitrary Archimedean vectorlattices. This survey describes the main properties of finite and totally finite elements in arbitrary andnormed vector lattices and in their sublattices. For most ofthe classical vector lattices the collection ofall finite elements isindicated. Finite and totally finite elements and their relations to finite functions arestudied in vector lattices of continuous functions. In special vector lattices of regular operators the firstresults on their finite elements will be presented. Then there is investigated the interesting question on thebehaviour of finite elements, when the vector lattice is represented (isomorphically) as a vector lattice ofcontinuous functions on a locally compact space, where one expects that finite elements are mapped intofinite functions. Special situations are discussed where this is the case. By making use of the space ofmaximal ideals, in particular, a topological characterization of finite and totally finite elements by meansof compact subsets is given.

Key words: Vector lattice, Dedekind completeness, Finite element, Maximal ideals, Hull-kernel topol-ogy, Representation of vector lattices, Banach lattice, Regular operator, Modulus of an operator, Finiterank operator

MSC 2000: 46B42, 47B60, 47B65, 54A05

1 Introduction

Let S be a locally compact (noncompact) Hausdorff space andX(S) a vector lattice of con-tinuous functions onS, i.e.X(S) ⊂ C(S). Functions with a compact support are of specialinterest and one might ask for some vector lattice characterization of such a functionϕ. Thisis easy to do for a positive function: the family of the infima of all multiples ofϕ with anypositive functionx ∈ X(S) should be majorized by one and the same function, of course with aconstant depending onx (see figure 1). In general, the moduli of the functions have tobe used.In the sequel the abstract version of this description leadsto the Definition 1 of a finite element,already in an arbitrary Archimedean vector lattice. If the majorizing element itself is a finiteelement thenϕ is called totally finite, see Definition 2.

The notion of a finite and a totally finite element in (abstract) Archimedean vector latticesEwas introduced by B. M. Makarov and the author in 1973 (see [8]) and in 1975 (see [9]). Finiteand totally finite elements in vector lattices are thoroughly studied in a series of papers, see [8],[9], [10], [11], [3], [4], [5], [16].

In connection with finite and totally finite elements severalquestions are quite natural andthis survey will answer some of them, of course, under appropriate additional conditions and in

PROCEEDINGS Positivity IV - Theory and ApplicationsDresden (Germany), 155-172 (2006)

most cases in a condensed manner. To remove or weaken the latter seems to be desirable. Al-though as the rule, for the proofs we refer to the references it will be clear that special techniqueshave to be applied in order to establish the formula (3) contained in the main Definition 1, whenthe finiteness of some class of elements has to be proved. Moreover, some analysis of the for-mula (3) will help to derive more information about the structure of the finite elements in manyspecial cases. Only some typical proofs (Theorems 4, 5, 10 and 14) are provided in order todemonstrate these two ideas.

First of all, the relations of finite and totally finite elements in vector subspaces of a givenvector lattice are of interest (§2). As the rule, an additional structure of the vector lattice willgive some, and sometimes even exhaustive, information on the finite elements. Another cycle ofproblems is the study of both finite functions and finite elements in vector lattices of continuousfunctions on a locally compact Hausdorff space (§2.2), where one expects very close relationsbetween. Although, in general, both notions are different avery mild condition ensures at leastthat any finite function is a finite element. However it is hardfor a finite element to be a finitefunction.

If E is a Banach (or vector) lattice andH is a vector sublattice ofE then it is a naturalquestion to establish the relations betweenΦ1(E) andΦ1(H), i.e. we ask, whether (or underwhich conditions) do the following relations hold ?

a) Φ1(H) ⊂ Φ1(E), b) Φ1(E) ∩H ⊂ Φ1(H), c) Φ1(E) ∩H = Φ1(H). (1)

In §3 we present some first results on finite elements in vector lattices of regular operators.It turns out that the finiteness of a finite rank operator is closely related to the finiteness of allelements constituting such an operator. These investigations have to be understood as a startingpoint of a systematic study of finite elements in particular vector lattices of operators.

§4 is devoted to the representation theory of vector latticescontaining finite elements as vec-tor lattices of continuous functions. It is very natural, atleast under some conditions, to expectthat finite elements are isomorphically represented as finite functions. An Archimedean vectorlattice possessing a sufficient number of finite elements allows (under some additional condi-tions) a representation as a vector lattice of (everywhere finite-valued) continuous functions ona locally compactσ-compact Hausdorff space such that all finite elements are represented asfinite functions.

Finally, for an Archimedean vector lattice we study the space of maximal ideals equippedwith the hull-kernel topology. Since representations of vector lattices are actually constructedon topological spaces that are homeomorphic to subsets of the space of maximal ideals of thevector lattice (§5), one might expect to obtain some further information on finite elements bymore detailed investigation of that topological space. This gives the possibility for an abstractcharacterization of finite and totally finite elements by means of the compactness of a specialsubsets in this space which can be assigned to each element.

Recall some definitions, notations and elementary facts in an Archimedean vector lattice(E,E+) which will be used further on, where in the most cases we refer to [1] and [12].

• A setB ⊂ E is called aband if it is an order closed ideal, that is the limit (inE) of anyorder convergent net of the idealB belongs toB.

• Two elementsx, y ∈ E are calleddisjoint written asx ⊥ y, if |x| ∧ |y| = 0.

156 M. R. Weber

• For any nonempty subsetA ⊂ E denote byAd the set{x ∈ E : x ⊥ y for any y ∈ A}.The setAdd is known as theband generated byA, the smallest band that containsA. If Aconsists of one single elementx, the band generated by{x} is called aprinciple band.

• A bandB in E is aprojection bandif E = B ⊕ Bd. In this case any elementx ∈ E hasa unique representationx = x1 + x2, wherex1 ∈ B andx2 ∈ Bd. The mapPB : E → E

defined byPB(x) = x1 for anyx ∈ E = B ⊕ Bd is a positive projection. In a Dedekindcomplete vector lattice any band is a projection band.

• If {u}dd is a projection band thenP{u} is denoted byPu. In this case for each elementx ≥ 0 there exists the elementsup{x ∧ n|u|} and,Pu(x) (for x ≥ 0) is calculated by theformula

Pu(x) = sup{x ∧ n|u|} . (2)

• A vector latticeE is said to have theprincipal projection property(PPP), if{x}dd is aprojection band for eachx ∈ E. Any σ-Dedekind complete vector lattice has the (PPP).

• An elementu ∈ E+ is aorder unit, if for eachx ∈ E there is aλ ∈ R with−λu ≤ x ≤ λu

(or equivalently,|x| ≤ λu).

• An elemente ∈ E+ is aweak order unit, if x ∈ E andx ⊥ e imply x = 0, i.e.{e}dd= E.

• An element0 < a ∈ E is called anatom of E, if whenever0 ≤ b ≤ a one hasb = λa.A Banach lattice is said to beatomic if for eachx > 0 there is an atoma such that0 < a ≤ x.

• A vector latticeE not possessing any order unit is called oftype (Σ) if E contains asequence of elements(en)

∞

n=1 with the following property

(Σ′)

{e1 ≤ e2 ≤ · · · ≤ en ≤ · · · ,for anyx ∈ E there existn ∈ N andC > 0 such that|x| ≤ Cen .

2 Finite and totally finite elements in arbitrary Archimedeanvector lattices

2.1 Definition of finite and totally finite elements

LetE be an Archimedean vector lattice.

Definition 1. An elementϕ ∈ E is calledfinite, if there is an elementz ∈ E satisfyingthe following condition: for any elementx ∈ E there exists a numberCx > 0 such that theinequality

|x| ∧ n|ϕ| ≤ Cxz (3)

holds for alln ∈ N.

Finite Elements in Vector Lattices 157

The elementz is called anE-majorantor briefly amajorantof the finite elementϕ, seefigure 1. The set of all finite elements of a vector latticeE is denoted byΦ1(E). Obviously,Φ1(E) is an ideal, i.e. a solid (sometimes also called normal) linear subspace ofE. The trivialcases forΦ1(E) to be even a projection band inE areΦ1(E) = E andΦ1(E) = {0}. Thegeneral case is considered in [4], Thm.2.8:

Theorem 1. The idealΦ1(E) is a projection band of the vector latticeE if and only ifE =

E1 ⊕E0, whereΦ1(E1) = E1 andΦ1(E0) = {0}. In this caseE1 = Φ1(E).

ϕ

x

ϕ

nϕ

zxc z

x n∧ ϕ

x

x n∧ ϕϕ

nϕ

Figure 1: Finite elementϕ with majorantz

The special class of finite elements characterized by possessing at least oneE-majorant,which itself is a finite element, in general, turns out to be different fromΦ1(E).

Definition 2. A finite elementϕ ∈ E is called totally finite if it has anE-majorant zbelonging toΦ1(E).

The set of all totally finite elements of a vector latticeE is also an ideal which will bedenoted byΦ2(E). Obviously, there hold the inclusions{0} ⊂ Φ2(E) ⊂ Φ1(E) ⊂ E, whichmight be proper (see [16]):

{0} ⊆ Φ2(E) ⊆ Φ1(E) ⊆ E possible case

6= 6= 6= yes

= = 6= yes

6= = 6= yes

6= = = yes

6= 6= = no

= 6= = no

= 6= 6= no

= = = E = {0}.

In [9] it was shown that any elementϕ ∈ Φ2(E) possesses anE-majorant which itself is atotally finite element, see also§ 5. It is clear thatΦ1(E) = E impliesΦ2(E) = Φ1(E).Each atom in a vector lattice is a totally finite element with itself as a majorant ([3], Prop.2.2).

158 M. R. Weber

It is easy to show thatΦ1(E) = Φ2(E) = E for the vector latticeE = c00 of all real sequenceswith finite support and for the vector latticeE = K(S) of all finite (i.e. with a compact support)continuous functions on a locally compact topological Hausdorff spaceS, see also the vectorlattice in Corollary 6. Below (§2.4) we will detect vector latticesE with Φ1(E) = {0}.

From the definitions one has

Theorem 2. If a vector latticeE has an order unit, thenΦ1(E) = Φ2(E) = E.

As a consequence, for the following classical vector lattices one immediately obtains thatany element is finite:If E is one of the vector latticesc, l∞, L∞

(µ) or C(K) (whereK is a compact Hausdorffspace), thenΦ1(E) = Φ2(E) = E.

Example for a finite element not being totally finite.Let beT = [−2, 2] \ {1, 1

2, 1

3, 1

4, . . .}. The required vector lattice will be constructed by the help

of the following functions

eν(t) =

ν∑

k=1

1

|kt− 1|, (t ∈ T ) ν = 1, 2, . . . .

The vector latticeE consists of all functionsf on [−2, 2] restricted toT such that

– f is continuous on[−2, 2] except a finite number of points1n

– for anyn there exists the finite limitlimt→

1

n

f(t)|nt− 1|.

ThenE is vector lattice (even a uniformly complete and of type(Σ)) with the propertyΦ1(E) 6=

Φ2(E), since one has (see [10])(i) If ϕ ∈ E and∃δ > 0 such thatϕ(t) = 0 for all t ∈ T ∩ [0, δ), thenϕ ∈ Φ1(E).

(ii) E ∋ ψ is totally finite (i. e.ψ ∈ Φ2(E)) if and only if ∃δ > 0 such thatψ(t) = 0 for allt ∈ T ∩ (−δ, δ).

2.2 Finite elements and finite functions in vector lattices of continuousfunctions

LetE(S) be vector lattice of continuous functions on some locally compact Hausdorff spaceS,i.e.E(S) ⊂ C(S).The finite functions inE(S) areK(S)∩E(S). As usual the finite elements inE(S) are denotedby Φ1(E(S)). If E(S) = C(S) thenΦ1(C(S)) = K(S). In general,

a) K(S) ∩E(S) * Φ1(E(S)) and b) Φ1(E(S)) * K(S) ∩ E(S).

An example for case a) is the vector latticeE of all continuous functions on[0,∞) vanishing at0. The finite function

ϕ(t) =

t, if t ∈ [0, 1]

2 − t, if t ∈ (1, 2]

0, if t ∈ (2,∞)

Finite Elements in Vector Lattices 159

belongs toE but fails to be a finite element. Indeed, ifz would be anE-majorant ofϕ then|x| ∧ nϕ ≤ cxz for somecx > 0 and anyn ∈ N. The functionx0(t) =

√z(t) belongs also to

E and with somec0 > 0 one has√

z ∧ nϕ ≤ c0z for all n ∈ N. Sinceϕ(t) > 0 on (0, 1) onehas(

√

z ∧ nϕ) (t) =√z(t) and so

√z(t) ≤ c0z(t) on (0, 1). This implies0 < 1

c20

≤ z(t) whatcontradicts toz(t)−→

t→00.

An example for case b) provides the vector latticeE of all continuous functionsx on[1,∞) × [1,∞) such that there existn ∈ N andλ > 0 with the property|x(t, s)| ≤ λtn

for ∀(t, s) ∈ [1,∞)× [1,∞). In this vector lattice an elementx belongs toΦ1(E) if and only ifx(t, s) = 0 on a set[a,∞) × [1,∞), where0 < a = a(x). Clearly, not all such functions havea compact support.

The following conditions on a vector latticesE(S) ⊂ C(S) turn out to be very importantnot only for a detailed investigation of them but also for therepresentation theory of generalvector lattices containing finite elements:Condition (⋆): For∀s ∈ S there∃x ∈ E(S) such thatx(s) 6= 0.Condition (Φ): Any finite element ofE(S) is a finite function.Condition (α): For any ordered pair of pointss0, s1 ∈ S (s0 6= s1) there∃ a finite function

x ∈ E(S) with x(sk) = k, k = 0, 1.The condition (⋆) avoids the case a), i.e. there holds: If a vector latticeE(S) of continuousfunctions on a locally compact Hausdorff spaceS satisfies the condition(⋆), then any finitefunction ofE(S) is a finite element of the vector latticeE(S). Obviously, the condition (Φ)avoids the case b).

In [8] there are proved the following properties for a vectorlatticeE(S) which satisfies thecondition(α):

1) If K ⊂ S is compact ands0 /∈ K then∃ a finite functionx0 ∈ E(S) such thatx0(s0) = 0

andx0(s) ≥ 1 onK

2) If F ⊂ S is closed ands1 /∈ F then∃ a finite functionx ∈ E(S) such thatx(s1) = 1 andx(s) = 0 onF

3) LetE(S) be uniformly complete. Then together with any finite function x0 ∈ E(S) thevector latticeE(S) contains all finite functionsx ∈ C(S) such thatsupp(x) ⊂ supp(x0)

4) Let f be any discrete linear functional onE(S), i. e. f(x ∨ y) = max{f(x), f(y)}. If fdoes not vanish onK(S) ∩E(S) thenf = c δs for some points ∈ S andc ∈ R+

5)Let S be locally compact,σ-compact. Letthe vector latticeE(S) be uniformly com-plete, of type(Σ) and additionally satisfythe conditions(Φ) and(α)

=⇒

For any discrete linear functionalfthere∃ a finite elementx0 ∈ E(S)

such thatf(x0) 6= 0.(Thenf = c δs also holds for somes ∈ S andc ∈ R+)

2.3 Finite elements in arbitrary vector lattices

We continue the study of finite and totally finite elements in vector lattices. For a given vectorsublatticeX of a vector latticeE an elementz ∈ E+ is called ageneralized order unitfor X if

160 M. R. Weber

for eachx ∈ X there is a real numberCx > 0 with |x| ≤ Cxz. Note thatE belongs then to theideal generated inE by z and thatz is not required to belong toX+ = X ∩E+.

Theorem 3. LetE be a vector lattice. Ifϕ ∈ E is a finite element then{ϕ}dd has a generalizedorder unit and{ϕ}dd

⊂ Φ1(E).

The converse statement of Theorem 2 is also true ifE contains a weak order unit.

Corollary 1. LetE be a vector lattice with a weak order unit. ThenΦ1(E) = Φ2(E) = E

if and only ifE has an order unit.

A weak order unit of a vector latticeE fails to be an order unit in general, even ifΦ1(E) =

Φ2(E) = E andE has an order unit. For example,u = (1, 12, . . . , 1

n, . . .) is a weak order unit

but not an order unit in the vector latticeE = c.In a vector lattices with (PPP) the finite elements can be characterized as follows.

Theorem 4. LetE be a vector lattice with (PPP). Then for an elementϕ ∈ E the followingstatements are equivalent:

1) ϕ is a finite element ofE.

2) {ϕ}dd has a generalized order unitz ∈ E+.

3) {ϕ}dd has an order unitz0 ∈ {ϕ}dd.

Proof.2) ⇒ 3). If z ∈ E+ is a generalized order unit of{ϕ}dd then for eachx ∈ {ϕ}dd,there is a real positive numberCx such that|x| ≤ Cxz. Let Pϕ be the band projection fromEonto{ϕ}dd. Then|x| = Pϕ|x| ≤ Pϕ(Cxz) = CxPϕz = Cxz0, wherez0 = Pϕz ∈ {ϕ}dd. Thisimplies thatz0 is an order unit of{ϕ}dd.3) ⇒ 1). SincePϕ|x| ∈ {ϕ}dd for arbitraryx ∈ E there is a positive numberCx such thatPϕ|x| ≤ Cxz0. Then by using the formula (2) one has

|x| ∧ n|ϕ| ≤ sup{|x| ∧ n|ϕ|} = Pϕ|x| ≤ Cxz0 for all n ∈ N.

This implies thatϕ is a finite element ofE.1) ⇒ 2) is precisely Theorem 3.

As a corollary one obtains

Corollary 2. LetE be a vector latticeE with (PPP). ThenΦ1(E) = Φ2(E) andΦ1(E) hasthe (PPP).

Theorem 4 shows that the finiteness of an element in an Archimedean vector lattice can bedetected by the properties of the principle band it generates. Namely, letϕ ∈ E be such that{ϕ}dd is a projection band. Then the elementϕ is finite if and only if{ϕ}dd has an order unit.In particular, ifE is aσ-Dedekind complete vector lattice thenΦ1(E) = E if and only if eachprincipal band possesses an order unit. So, if for anσ-Dedekind complete vector latticeE onehasΦ1(E) 6= E then there exists at least one principal band without order unit.

Finite Elements in Vector Lattices 161

2.4 Finite elements in Banach lattices

Without (PPP), the structure of a Banach lattice helps to describe the finite elements.

Theorem 5. LetE be a Banach lattice andϕ ∈ E. Then the following statements are equiva-lent:

1) ϕ is a finite element.

2) The closed unit ballB({ϕ}dd) of {ϕ}dd is order bounded inE.

3) {ϕ}dd has a generalized order unit.

Proof.We show only the equivalence of 1) and 3). For details see [3] and [4]. So, it is toshow that the elementϕ is finite, if {ϕ}dd has a generalized order unit. In fact, letz ∈ E+ be ageneralized order unit of{ϕ}dd. Define a norm on{ϕ}dd by

‖x‖z = inf{λ > 0: |x| ≤ λz}, x ∈ {ϕ}dd.

Then by Theorem 12.20, [1], the space({ϕ}dd, ‖ · ‖z

)is anAM-space, where|x| ≤ ‖x‖zz

holds. Since the band{ϕ}dd is closed inE ([12], Prop.1.2.3)({ϕ}dd, ‖ · ‖) also is a Banachspace. The open mapping theorem implies that the norms‖·‖ and‖·‖z are equivalent on{ϕ}dd.In particular, there is aC > 0 such that‖x‖z ≤ C‖x‖ for all x ∈ {ϕ}dd. Now |x| ≤ ‖x‖zz foreachx ∈ {ϕ}dd, implies‖x‖z ≤ C, i.e. |x| ≤ Cz, for eachx ∈ {ϕ}dd with ‖x‖ ≤ 1. If x ∈ E

is now an arbitrary element then0 ≤ |x| ∧ n|ϕ| ≤ |x| (and hence‖|x| ∧ n|ϕ|‖ ≤ ‖x‖) implies|x| ∧ n|ϕ| ≤ ‖x‖Cz for all n ∈ N, which means thatϕ is finite.

The principal band generated by a finite element may fail to possess an order unit as thefollowing example shows. LetE = C[0, 1] andH = {x ∈ E : x(t) = 0, ∀t ∈ [0, 1

2]}. Then

Φ1(E) = E, the idealH is a principal band, moreover,H = {ϕ}dd for anyϕ ∈ H satisfyingϕ(t) 6= 0 for t ∈ (

12, 1] andH does not possess any order unit. However, each functionz ∈ E

with z(t) > 0 for t ∈ (12− δ, 1] is a generalized order unit, whereδ is some positive sufficiently

small number. For details see [3].For a Banach latticeE denote byΓE the set of all atoms ofE with norm1. As was men-

tioned after Definition 2 the inclusionΓE ⊂ Φ1(E) holds.

Theorem 6. Let the norm of the Banach latticeE be order continuous. Then1) Φ1(E) = Φ2(E) = span(ΓE)

1,

2) Φ1(E) is closed inE if and only ifΓE is a finite set, particularly,Φ1(E) = E if and onlyif E is finite dimensional.

For the following classical vector lattices one immediately obtains the following informa-tion on the finite elements:a) If E is one of the vector latticesc0 or lp with 1 ≤ p < ∞ then Φ1(E) = Φ2(E) =

span(ΓE) = span{ek : k = 1, 2, . . .}, whereek ∈ E is the sequence whichk′s term equals1 and all others are0,b) If E = Lp

(a, b) with 1 ≤ p <∞ thenΦ1(E) = {0}.

The vector lattice of all finite continuous functions onR is an example of a vector latticepossessing the propertyΦ1(E) = Φ2(E) = E. In the next theorem the class of Banach latticeswith this property is characterized.

1If ΓE = ∅ then we definespan(ΓE) = {0}.

162 M. R. Weber

Theorem 7 (Characterization of Banach lattices withΦ1(E) = Φ2(E) = E).

For a Banach latticeE the following statements are equivalent:

1) Φ1(E) = Φ2(E) = E.

2) E is lattice isomorphic to anAM-space and each pricipal band has a generalized orderunit.

Another result for a vector latticeE to satisfyΦ1(E) = E uses the structure of a strictinductive limit (see [8], Thm.5.3).

A vector lattice is calledof type(LF ), if it is the strict inductive limit of an increasingsequence ofF -spaces, where the topology is defined by a sequence of monotone seminorms.

Theorem 8. LetE be a vector lattice of type(Σ) and of type(LF ). ThenE = Φ1(E).

2.5 Finite elements in sublattices

We start with three natural situations, where a given vectorlattice is embedded into anothervector lattice.

If E denotes the Dedekind completion of an Archimedean vector latticeE thenΦ1(E) =

Φ1(E) ∩ E, i.e. the relation c) of (1) is true. This follows from a general result which can beproved for any majorizing vector sublatticeH of a vector latticeE (see [4], Thm.2.3).

If E denotes the norm completion of a normed vetor latticeE thenE is norm dense inE,however, in general,Φ1(E) ⊂ Φ1(E) does not hold. Whether the inclusionΦ1(E) ∩ E ⊂

Φ1(E) is true or not, is not known. The norm completion of the vectorlattice c00 equippedwith the supremum norm is the Banach latticec0. In this caseΦ1(c00) = c00 = Φ1(c0). Onthe other hand the Banach latticeL1

(0, 1) is the norm completion of the vector latticeC[0, 1]

equipped with the integral norm induced fromL1(0, 1). In this caseΦ1(C[0, 1]) = C[0, 1] and

Φ1(L1(0, 1)) = {0}.

If E is a Banach lattice,E ′′ its bidual andj : E → E ′′ the canonical embedding, thenj(Φ1(E)) ⊂ Φ1(E

′′). After identification ofj(E) with E this result can be written asΦ1(E) ⊂

Φ1(E′′) ∩ E. The equation however is not true, in general, as the vector latticec0 shows. For

the details see [4], Thm.2.10.

Observe that the inclusion a) of (1) may not hold ifH is an arbitrary ideal of a Banach latticeE or if H is a norm closed sublattice which is the range of a positive projection onE. It holdsif H is a closed ideal.The inclusion b) of (1) may not hold ifH is a closed ideal or a band inE. It holds ifH is avector sublattice which is the range of a positive projection.The relation c) (and therefore also a) and b)) holds ifH is a projection band:

Theorem 9. If H is a projection band in a vector latticeE andPH the band projection fromEontoH, thenPH

(Φ1(E)

)= Φ1(E) ∩H = Φ1(H).

For an example and also for a description of the finite elements in the direct sumsc0(I, Ei),

ℓp(I, Ei) for p ∈ [1,∞) andℓ∞(I, Ei) of Banach latticesEi, whereI is an arbitrary index set,we refer to [2], [4].

Finite Elements in Vector Lattices 163

The indicated spaces are defined as follows. Let bex = (xi)i∈I , xi ∈ Ei. Then

x = (xi)i∈I ∈

c0(I, Ei), if ∀ε > 0 ∃ finite setIε ⊂ I with ‖xi‖ < ε, ∀i /∈ Iε

ℓp(I, Ei), if∑i∈I

‖xi‖p <∞

ℓ∞(I, Ei), if supi∈I

‖xi‖ <∞ .

The linear operations and the order are understood to be the point- or coordinatewise ones, thenorms are defined by

‖x‖ = ‖(xi)i∈I‖ =

supi∈I

‖xi‖, if x ∈ c0(I, Ei), ℓ∞(I, Ei)

(∑i∈I

‖xi‖p

) 1

p

, if x ∈ ℓp(I, Ei), 1 ≤ p <∞

An element(ϕi)i∈I belongs toΦ1(c0(I, Ei)) andΦ1(ℓp(I, Ei)) if and only if eachϕi ∈

Φ1(Ei) for all i ∈ I andϕi = 0 for all but finite manyi ∈ I. An element(ϕi)i∈I is inΦ1(ℓ∞(I, Ei)) if and only if ϕi ∈ Φ1(Ei), ∀i ∈ I and there exist0 ≤ zi ∈ Ei such that eachclosed unit ballB({ϕi}

dd) is a subset of[−zi, zi] andsup

i∈I

‖zi‖ <∞.

3 Finite elements in vector lattices of regular operators

3.1 Regular operators on Banach lattices

For two vector latticesE, F , whereF is Dedekind complete the vector latticeLr(E, F ) is

Dedekind complete. So, Corollary 2 impliesΦ1(Lr(E, F )) = Φ2(L

r(E, F )).

A consequence of Theorem 4 is that

Orth(E) = {I}dd= {T ∈ L

r(E) : − λI ≤ T ≤ λI} ⊂ Φ1(L

r(E)) = Φ2(L

r(E)),

in particular, the identity operatorIE is a finite element inLr(E) with itself as anLr

(E)-majorant, provided the Banach latticeE is Dedekind complete. That is for anyU ∈ L

r(E)

there exists a positive numbercU such that

|U | ∧ nIE ≤ cUIE for all n ∈ N . (4)

Theorem 10. Let E and F be Banach lattices such thatF is Dedekind complete and letT : E → F be a lattice isomorphism (ontoF ). ThenT is a finite element inLr

(E, F ). More-over,T is a majorant of itself.

Proof.SinceT is a lattice isomorphism,E is Dedekind complete as well. The order continuityof T (as any lattice isomorphism) guarantees the equalities

|TU | = T |U | and T (U1 ∧ U2) = (TU1) ∧ (TU2) for any U, U1, U2 ∈ Lr(E). (5)

If S ∈ Lr(E, F ) then, obviously,T−1S ∈ L

r(E) and therefore (4) implies

|T−1S| ∧ nIE ≤ cT−1SIE , for n ∈ N. (6)

164 M. R. Weber

If now the (positive) operatorT is applied to the inequality (6), then by means of (5)

T(|T−1S| ∧ nIE

)= |S| ∧ nT ≤ cT−1ST for n ∈ N

follows, which shows that the operatorT is a finite element inLr(E, F ) with itself as a majo-

rant.An application of techniques and results from§2 is now

Theorem 11. If E andF be Banach lattices such thatF is Dedekind complete. Let beH aband ofF andP : F → H the band projection. Then

1) Lr(E,H) is a projection band ofLr

(E, F ).

2) Φ1(Lr(E,H)) = Φ1(L

r(E, F )) ∩ L

r(E,H) = {PT : T ∈ Φ1(L

r(E, F ))}.

For Banach latticesE andF define the mapping

P : Lr(E, F ′′

) → Lr(F ′, E′

)

by P(T ) = T ′J for T ∈ Lr(E, F ′′

), whereT ′: F ′′′

→ E ′ is the adjoint operator toT andJ : F ′

→ F ′′′ the canonical embedding2. The following fact is established in [6], Thm.5.6 andwill be used to prove the next theorem:P is an isometric lattice isomorphism from(L

r(E, F ′′

), ‖ · ‖r

)onto

(L

r(F ′, E′

), ‖ · ‖r

), where‖ · ‖r denotes the regular operator norm,

respectively.

Theorem 12. LetE andF be Banach lattices, andP as above. DenoteA = Lr(E, F ′′

) andB = L

r(F ′, E′

). Then

1) T ∈ A is finite if and only ifP(T ) is finite inB, i.e.P (Φ1(A)) = Φ1(B).

2) U ∈ L+(E, F ′′) is anA-majorant ofT if and only ifP(U) is aB-majorant ofP(T ).

Note that ifT ∈ Lr(E, F ) thenjT ∈ A, where againj : F → F ′′ denotes the canonical

inclusion mapping. SinceT ′= P(jT ) we have

Corollary 3. Let E and F be Banach lattices andT ∈ Lr(E, F ). ThenT ′ is finite in

Lr(F ′, E′

) if and only ifjT is finite inLr(E, F ′′

).

As a special case, whenF is a reflexive Banach lattice, we obtain that the finiteness ofanoperatorT : E → F can be characterized by the finiteness of its adjointT ′.

Corollary 4. If F is a reflexive Banach lattice then for each Banach latticeE, an operatorT ∈ L

r(E, F ) is a finite element if and only ifT ′ is finite inLr

(F ′, E′), i. e.

Φ1

(L

r(F ′, E′

))

={T ′

: T ∈ Φ1

(L

r(E, F )

)}.

Moreover,U ∈ L+(E, F ) is anLr(E, F )-majorant ofT if and only ifU ′ is anL

r(F ′, E′

)-majorant ofT ′.

For finite elements in the vector lattice of regular operators defined on anAL-space there aresome results gathered in the next theorem ([5], Thm.2.7), where the vector latticeF is alwaysassumed to be Dedekind complete.

2where the notationJ is chosen only for stressing the slightly different situation.

Finite Elements in Vector Lattices 165

Theorem 13 (Regular operators onAL-spaces).

1) E is anAL-space andF is anAM-space with order unit =⇒ Φ1(L

r(E, F )) = L

r(E, F ).

2) E is anAL-space andF is anAM-space ⇐= Φ1(L

r(E, F )) = L

r(E, F ).

3) LetE be a Dedekind complete Banach lattice.ThenΦ1

(L

r(E)

)= L

r(E) if and only ifdim E <∞.

The Banach latticeF may fail to have an order unit even ifΦ1

(L

r(E, F )

)= L

r(E, F ), the

Banach latticeE is anAL-space andF is anAM-space. For an example see [5] and [3].

3.2 Finite rank operators in Lr(E, F )

Now we consider the finite rank operators inLr(E, F ), whereE, F are vector lattices andF is

Dedekind complete.Let beψ′

1, . . . , ψ′

n∈ E ′ andϕ1, . . . , ϕn ∈ F andT =

n∑i=1

ψ′

i⊗ ϕi. Each finite rank operator

possesses a compact modulus (see [1], Thm.5.7), which is dominated by the operator

n∑

i=1

|ψ′

i| ⊗ |ϕi|

however, need not coincide with it.In [5] §4 there is proved the following result.

Theorem 14 (Finite rank operators).

LetT =

n∑i=1

ψ′

i⊗ ϕi belong toLr

(E, F ). Let at least eitherψ′

1, . . . , ψ′

nbe pairwise disjoint in

E ′ or ϕ1, . . . , ϕn be pairwise disjoint inF . Then

T ∈ Φ1(Lr(E, F )) ⇐⇒ ψ′

i∈ Φ1(E

′) andϕi ∈ Φ1(F ), i = 1, . . . , n.

For rank one operators we get thatψ′⊗ ϕ is a finite element inLr

(E, F ) if and only ifψ′

∈ Φ1(E′) andϕ ∈ Φ1(F ).

In order to prove at least one special case of the formulated theorem and also to demonstratehow from the defining inequality (3) of a finite element one canderive additional informationon the structure of the finite elements in some given vector lattice we show thatψ′

∈ Φ1(E′)

andϕ ∈ Φ1(F ) provided the rank one operatorT = ψ′⊗ ϕ is a finite element inLr

(E, F )

with a majorantU ∈ L+(E, F ). Namely, without loss of generality we may assumeψ′≥ 0 and

ϕ ≥ 0. Then for eachS ∈ Lr(E, F ) there is a positive constantcS such that

|S| ∧ nT ≤ cSU for all n ∈ N. (7)

166 M. R. Weber

In particular, ifS = ψ′⊗ h ∈ L

r(E, F ), whereh ∈ F then it follows from (7) and Theorem

1.16 of [1] that

cSUx ≥

(|S| ∧ nT

)(x)

= inf

{ k∑

i=1

(|S|xi) ∧ (nTxi) : xi ∈ E+,

k∑

i=1

xi = x, n ∈ N

}

= inf

{ k∑

i=1

(ψ′(xi)|h|) ∧ (ψ′

(xi)nϕ) : xi ∈ E+,

k∑

i=1

xi = x, n ∈ N

}

= inf

{ k∑

i=1

ψ′(xi)(|h| ∧ (nϕ)) : xi ∈ E+,

k∑

i=1

xi = x, n ∈ N

}

= ψ′(x)(|h| ∧ (nϕ))

for all x ∈ E+ andn ∈ N. One finds somex0 ∈ E+ such thatψ′(x0) = 1 and has

|h| ∧ (nϕ) ≤ cSUx0 for n ∈ N,

which implies thatϕ is a finite element ofF .On the other hand, ifS = h′ ⊗ ϕ, whereh′ ∈ E ′ then again by means of (7) for arbitary

x, xi ∈ E+ withk∑

i=1

xi = x one has

(|h′| ∧ nψ′

)(xi)ϕ ≤

(|h′|(xi)ϕ

)∧

(nψ′

(xi)ϕ)

= (|S|xi) ∧ (nTxi)

for each1 ≤ i ≤ k, and thus(|h′| ∧nψ′)(x)ϕ ≤

k∑i=1

(|S|xi)∧ (nTxi). It follows from Theorem

1.16 of [1] that(|h′| ∧ nψ′) (x)ϕ ≤ (|S| ∧ (nT )) (x) ≤ cSUx for all x ∈ E+ andn ∈ N. With

someu′ ∈ F ′

+ such thatu′(ϕ) = 1 one has

(|h′| ∧ nψ′) (x) ≤ u′(cSUx) = cS(U ′u′)(x) x ∈ E+, n ∈ N.

Therefore

|h′| ∧ nψ′≤ cS(U ′u′) = cSv

′ n ∈ N,

wherev′ = U ′u′ ∈ E ′

+. This shows thatψ′ is finite inE ′.

4 Representation of vector lattices containing finite elements

Let E be an Archimedean vector lattice andS a topological Hausdorff space. A vector latticeE(S) ⊂ C(S) is called arepresentation3 of the vector latticeE, if there is a vector latticeisomorphismi : E → E(S). In many situations representations of Archimedean vectorlattices(Dedekind complete or not) are considered preferably on compact (extremally disconnected)

3More exactly, the representation ofE should be understood as the pair (E(S), i).

Finite Elements in Vector Lattices 167

spacesS, where sometimes continuous functions are allowed, that might take on even infinitevalues on nowhere dense subsets ofS, see e.g. [13].

For Archimedean vector lattices containing nontrivial finite elements it seems to be nat-ural if one looks for representations consisting of continuous functions on a locally compactspace. This would give the possibility to represent finite elements of the vector lattice as finitefunctions.

A representationE(S) is called a (⋆)-representation, if the vector latticeE(S) satisfies thecondition (⋆), see§ 2.2. Analogously, (Φ)- and (α)- and also (Φα)-representations are defined.Based on the S.Kakutani-H.F.Bohnenblust-M.G.Krein-S.G.Krein-Theorem ([1], Thm.12.28) avector latticeE of type (Σ) possesses a (⋆)-representation consisting of bounded functions ifthere exists a monotone norm onE, ([8], Thm.1.3). A vector lattice of type (Σ) can not have a(⋆)-representation on a compact spaceS ([15], Lemma 7).

The next results show that for vector lattices of type (Σ) there exist more qualified rep-resentations such that any member of a given countable collection of finite elements will bemapped by means of an isomorphism on a finite function. For thedetails (and other kinds ofrepresentations) see [14], [8], [15], [11].

Theorem 15. Let E be a vector lattice of type(Σ). Let E admit a monotone norm and let{ϕn}n∈N be a sequence of finite elements ofE.Then there exists a(⋆)-representation on someσ-compact (locally compact) spaceS such thateach elementϕn, n ∈ N is represented as a finite functions onS.

Theorem 16. LetE be a vector lattice of type(Σ).Then forE to possess a(Φα)-representation on someσ-compact spaceS it is sufficient and,in case of the uniform completeness ofE, also necessary, that there exists a sequence of finiteelements{ϕn}n∈N in E satisfying the condition: for any discrete(see§2.2) functionalf thereexists a numbern such thatf(ϕn) 6= 0.

Corollary 5. LetE be a vector lattice of type(Σ) such thatE = Φ1(E).ThenE possesses a(Φα)-representationE(S) on someσ-compact spaceS.If moreover,E is uniformly complete, thenE(S) = K(S).

Together with Theorem 8 from Corollary 5 follows now

Corollary 6. LetE be a vector lattice of type(Σ) and of type(LF ). ThenE has a repre-sentation asK(S) on some locally compactσ-compact spaceS (see [7]).

Indeed, the vector latticeE is uniformly complete since it is the strict inductive limitof itssubspaces of type (F ). The equalityE = Φ1(E) is established in Theorem 8.

In the next section among others we study the relation between the spaceS of a represen-tationE(S) and the space of all maximal ideals ofE topologized in an appropriate manner,where the latter space or some of its subspaces turn out to be homeomorphic toS.

168 M. R. Weber

5 The space of maximal ideals of a vector lattice

For an Archimedean vector latticeE denote byM(E) the set of all maximal ideals ofE. Forany discrete functionalf 6= 0 onE one hasf−1

(0) ∈ M(E) and, vice versa, anyM ∈ M(E)

defines a discrete functional4 (up to a constant coefficient).The collectionM(E) will be equipped with the hull-kernel topologyτhk by defining the

closure of any subsetA ⊂ M(E): a maximal idealM0 ∈ M(E) belongs to the closure ofA(i.e.M0 ∈ A), iff M0 ⊃

⋂M∈A

M .

If (E(S), i) is a (⋆)-representation ofE then for each points ∈ S the sets

Hs = {x ∈ E(S) : x(s) = 0}

and i−1(Hs) are maximal ideals inE(S) andE, respectively. The canonical mapκ : S →

M(E) defined byκ(s) = i−1(Hs) is always continuous. Under additional conditions the spaces

S andM(E) might be even canonically homeomorphic.Important subsets ofM(E) are obtained as follows. For anyx ∈ E put Gx = {M ∈

M(E) : x /∈ M}. Then the setsuppM(x) = Gx is called theabstract support of the elementx. ForA ⊂ E put G(A) =

⋃x∈A

Gx and use the special notationMΦ(E) = G(Φ1(E)

)for

A = Φ1(E).The main properties of the topologyτhk:

a) Gx is aτhk-open set for eachx ∈ E

b) The system{Gx}x∈E is a basis of the toplogyτhk

c)(M(E), τhk

)is a Hausdorff space

d) MΦ(E) is the largest locally compact subspace contained inM(E) and the system{Gϕ}ϕ∈Φ1(E) is a basis ofτhk in MΦ(E).

will be used both for a deeper study of finite elements in vector lattices (see [9], [10]) as well asfor further developing of the representation theory in [8] and [11].

Theorem 17 (Topological characterization of finite and totally finite elements).

For a radical-free Archimedean vector latticeE the following assertions hold:

1) ϕ ∈ Φ1(E) ⇐⇒ suppM

(ϕ) is a compact set with respect toτhk

2) z ∈ E is anE-majorant of the finite elementϕ⇐⇒ suppM(ϕ) ⊂ Gz

3) ψ ∈ Φ2(E) ⇐⇒ suppM

(ψ) is compact andsuppM

(ψ) ⊂ MΦ(E)

4) Φ2(E) = Φ1(Φ2(E)), i.e. each totally finite element has a totally finite majorant.

4Concerning discrete functionals or maximal ideals in the vector latticeE we assume not only the existence ofthem but in most cases also that there are sufficient many to separate the vectors ofE (i.e.f(x) = 0 for all discretefunctionalsf impliesx = 0), or equivalently, thatE is radical-free, i.e.R = R(E) =

⋂{M : M ∈ M(E)} = {0}.

For example, vector lattices of type (Σ) and vector lattices such thatΦ1(E) = E are radical-free. In case ofR(E) 6= {0} the already radical-free vector latticeE/R will be considered instead ofE, because the spacesM(E) andM(E/R) are then homeomorphic.

Finite Elements in Vector Lattices 169

Many information on the finite elements ofE are related to topological properties ofM(E)

and of the subspaceMΦ(E). The compactness ofM(E) e.g. implies the existence of an oderunit inE and thereforeΦ1(E) = E. Theσ-compactness (and non-compactness) ofMΦ(E) isnecessary and sufficient forΦ2(E) to be a vector lattice of type (Σ).

Especially for vector lattices of type (Σ) there is known quite a lot aboutMΦ(E). In thatcase its closedness implies itsσ-compactness and by the properties 2) and 3) of Theorem 17 theequalityΦ2(E) = Φ1(E). ForMΦ(E) = MΦ(E) in M(E) we mention the following result

Theorem 18. LetE be of type (Σ).

For the closedness ofMΦ(E) in M(E) it is necessary and in case of the uniform completenessofE also sufficient that there holds both conditions:

(i) Φ2(E) is a vector lattice of type (Σ) and

(ii) Φ1(E) = Φ2(E).

All the conditions in that theorem, i.e. type (Σ), uniform completeness, (i) and (ii) areessential, as exhaustive examples in [10] demonstrate.

Now coming back to the question of representation we are ableto add some further results,where the underlying space is homeomorphic to a subspace ofM(E).

Based on the next theorem more qualified representations of vector lattices containing suf-ficiently many finite elements can be constructed, see [11].

Theorem 19. Let (E(S), i) be an (α)-representation ofE andE0 = {x ∈ E : ix ∈ K(S)}.Then

1) E0 is an ideal ofE andE0 ⊂ Φ2(E),

2) E0 is order dense5 in E, i.e.x ⊥ y for all x ∈ E0 impliesy = 0,

3) E0 is embeddable into a vector lattice with order unit,

4) S is homeomorphic toG(E0) (and also toM(E0)).

An idealE0 ⊂ E is called anR-base(representation base), ifE0 satisfies the first threeconditions of Theorem 19.

Definition 3. A representation(E(S), i) ofE is called anrepresentation by means of theR-baseE0 if iE0 ⊂ K(S) (i.e. iE0 is a vector lattice of finite functions) and satisfies the condition(α).

Theorem 19 implies that a representation by means of anyR-baseE0 is always a represen-tation onG(E0) = M(E0).

Theorem 20 (Existence of base representations).LetE be a vector lattice andE0 anR-basein E. Then there exists a representation by means ofE0.If (E(S), i) is this representation theniE0 = K(S) if and only ifE0 is uniformly complete.

5see [17], Thm.23.3.

170 M. R. Weber

Analogously to Theorem 16 one has now

Theorem 21.For the existence of a (Φα)-representation (E(S), i) for a vector latticeE on aσ-compact spaceS it is sufficient and, ifE is of type (Σ) and uniformly complete, also necessarythat the spaceM(E) is locally compact andσ-compact.

References

[1] C.D. Aliprantis and O. Burkinshaw.Positive Operators. Academic Press, Inc., London,1985.

[2] Z.L. Chen. On week sequential precompactness in Banach lattices.Chinese J. Contempo-rary Math., 20 (4):477–486, 1999.

[3] Z.L. Chen and M.R. Weber. On finite elements in vector lattices and Banach lattices.Math. Nachrichten, 279, No.5-6, 495–501 (2006).

[4] Z.L. Chen and M.R. Weber. On finite elements in sublattices of Banach lattices.Math.Nachrichten, to appear

[5] Z.L. Chen and M.R. Weber. On finite elements in lattices ofregular operators.Prepr.,Techn. Univ. Dresden, No.MATH-AN-06-03, 2003. .

[6] Z.L. Chen and A.W. Wickstead. Some applications of Rademacher sequences in Banachlattices.Positivity, 2:171-191, 1998.

[7] I. Kawai. Locally convex lattices.J. Math. Soc. Japan, 9, no.3-4, 1957.

[8] B.M. Makarow and M. Weber.Uber die Realisierung von Vektorverbanden I. (Russian).Math. Nachrichten, 60:281–296, 1974.

[9] B.M. Makarow and M. Weber. Einige Untersuchungen des Raumes der maximalen Idealeeines Vektorverbandes mit Hilfe finiter Elemente I.Math. Nachrichten, 79:115–130, 1977.

[10] B.M. Makarow and M. Weber. Einige Untersuchungen des Raumes der maximalen Idealeeines Vektorverbandes mit Hilfe finiter Elemente II.Math. Nachrichten, 80:115–125,1977.

[11] B.M. Makarow and M. Weber.Uber die Realisierung von Vektorverbanden III.Math.Nachrichten, 68:7–14, 1978.

[12] P. Meyer-Nieberg.Banach Lattices. Springer-Verlag, Berlin, Heidelberg, New York, 1991.

[13] B.Z. Vulikh. Introduction to the Theory of Partially Ordered Spaces. Wolters-Nordhoff,Groningen, 1967.

[14] M. Weber. Uber eine Klasse von K-Linealen und ihre Realisierung.Wiss. Zeitschrift THKarl-Marx-Stadt, XIII, Heft 1: 159–171, 1971.

Finite Elements in Vector Lattices 171

[15] M. Weber. Uber die Realisierung von Vektorverbanden II.Math. Nachrichten, 65:165–177, 1975.

[16] M.R. Weber. On finite and totally finite elements in vector lattices.Analysis Mathematica,21:237–244, 1995.

[17] A.C. Zaanen.Introduction to Operator Theory in Riesz Spaces. Springer-Verlag, Berlin,Heidelberg, New York, 1997.

Martin R. WeberFachrichtung MathematikTechnische Universitat DresdenD - 01062 Dresdene-mail: martin.weber@tu-dresden.de

172 M. R. Weber