Linear operators on Abramovich--Wickstead type spaces Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2008, Том 10, Выпуск 3, С. 46-55

YffK 517.98

LINEAR OPERATORS ON ABRAMOVICH-WICKSTEAD TYPE SPACES

F. Polat

In this note, we define and investigate Abramovich-Wickstead type spaces the elements of which are the sums of continuous functions and discrete functions. We give an analytic representation of regular and order continuous regular operators on these spaces with values in a Dedekind complete vector lattice.

Mathematics Subject Classification (2000): 54C35, 46E40, 46G10. Key words: CD0(K)-spaces, quasiregular measure, regular operator.

CDo-type spaces were firstly introduced by Yu. A. Abramovich and A. W. Wickstead in [1] and [2] and further investigated by S. Alpay and Z. Ercan in [3]. CDo-type spaces deserve to be called Abramovich-Wickstead spaces, or briefly AW-space as in [4], since they mainly stem from the works of Yu. A. Abramovich and A. W. Wickstead. In this note we construct a new type AW-space and call it CD0 for the sake of convenience.

Throughout this note, the symbols Lr and Lrn denote the space of regular and order continuous regular operators respectively. For unexplained terminology about vector lattice theory, we refer to [5].

The first section is devoted to some introductory knowledge about vector-valued measures. This section will be useful in obtaining main results. For more detailed information about vector-valued measures, we refer to [6]. The second section of this note contains the definition of CD0-spaces. The third section is devoted to description of regular operators charactreziations about linear operators on CD0-space by means of vector measures and order summation. The fourth section contains two main results of the paper. In this section we are mainly interested in regular and order continuous regular operators on CD0-space with values in Dedekind complete vector lattices.

Consider a nonempty set K and a a-algebra A of the subsets of K. Let E be a Dedekind complete vector lattice. We shall call the mapping y : A ^ E a vector measure if y(0) = 0 and for every sequence (An) of pairwise disjoint sets An £ A the equality holds

We say that a measure y is 'positive and write y ^ 0 if y(A) ^ 0 for all A £ A. We denote the set of all order bounded E-valued measures on a a-algebra A by cab(K, A, E). If v £ cab(K, A, E) and t £ R, then we put by definition

1. Vector Measures

© 2008 Polat F.

(1) + v)(A) := ^(A) + v(A) (A e A),

(2) (t^)(A) := i^(A) (A e A),

(3) ^ ^ v ^ ^ - v ^ 0.

One can prove that cab(K, A, E) is a Dedekind complete vector lattice. In particular, every measure ^ : A ^ E has the positive part := ^ V 0 and the negative part : = (— = —^ A 0. It is easy to verify that

(A) = sup{^(A') : A' e A, A' c A} (A e A).

In the sequel, we shall consider special E-valued measures. Suppose that K is a compact topological space and A is the Borel a-algebra. A positive measure ^ : A ^ E is said to be regular if for every A e A we have

^(A) = inf{^(U) : A c U, U e Op(K)}

where Op(K) is the collection of all open subsets of K. If the latter condition is true only for closed A e A, then ^ is called quasiregular. Finally, an arbitrary measure ^ : A ^ E is said to be regular (quasiregular) if the positive measures and are regular (quasiregular). Let rca(K, E) and qca(K, E) be the sets of regular and quasiregular E-valued Borel measures respectively. It is seen from the definitions that rca(K, E) and qca(K, E) are vector sublattices in cab(K, A, E). Clearly, the supremum (infimum) of the increasing (decreasing) family of quasiregular measures bounded in cab(A, A, E) will also be quasiregular. The same holds for regular measures. Thus qca(K, E) and rca(K, E) are Dedekind complete vector lattices.

We now define the integral with respect to an arbitrary measure ^ e cab(K, A, E). Let us denote by St(K, A) the set of step functions ^ : K ^ R of the form ^ = ^akXAk, where Ai,..., An e A, ai,..., an e R, and xa is the characteristic function of a set A. Construct the operator I" : St(K, A) ^ E by putting

I" I akXAk := akMAk). \k=1 ) k=1

As it is seen I" is a linear operator. Moreover the following normative inequality holds

M/)| < ||fIUM(K) (/ e st(K, A)),

where ||/= sup^K 1/(k)|. The subspace St(K, A) is dense with respect to the norm in the space ¿^(K, A) of all bounded A-measurable functions on K. Therefore I" admits a unique linear extension by continuity to l^(K, A), with the above-mentioned normative inequality being preserved. In particular, if K is a compact space and A is the Borel a-algebra, then I"(/) is defined for every continuous function / e C(K). Note also that I" ^ 0 if and only if p ^ 0.

Finally, we give the following result of J. D. M. Wright [7] about analytical representation of linear operators which will play an important role to obtain main results of this note.

Theorem 1. Let K be a compact topological space and let E be an arbitrary Dedekind complete vector lattice. The mapping ^ ^ I" implements a linear and lattice isomorphism of Dedekind complete vector lattices qca(K, E) and Lr (C(K), E).

2. CD0 (K, E)-spaces

In this section we introduce a new class of Abramovich-Wickstead type spaces. We start with the following definition which contains the building blocks of this space.

Definition 2. For a compact space K and a relatively uniformly complete vector lattice E, we set

(1) C(K, E(e)) the space of all mappings from K into E(e) which are continuous in the sense of the norm || ■ ||e where E(e) denotes the ideal generated by e £ E + and

||u||e := inf{A > 0 : |u| < Ae| (u £ E(e)).

Then, we set

Cr(K, E) := J {C(K, E(e)) : e £ E+}

and call the elements of this set r-continuous or uniformly continuous functions on K.

It is clear that Cr (K, E) is contained in E), the space of order bounded functions

from K into E, since in E(e) norm boundedness coincides with order boundedness. Moreover, Cr (K, E) is a vector sublattice in E).

(2) Co(K, E(e)) the space of all mappings d from K into E(e) such that for all e > 0 the set {k £ K : ||d(k)||e ^ e} is finite. Then we set

Co(K, E) := J {co(K, E(e)) : e £ E +} .

It is clear that Co(K, E) is contained in E). Moreover, Co(K, E) is a vector sublattice

in MK,E).

Now we give the following theorem which will be useful in the sequel. Theorem 3. Let K be a compact space. For any f £ Cr (K, E) and e > 0 there exist e £ E + and finite collections ^i,..., ^>n £ C(K) and ei,..., en £ E such that

sup

aSK

f (a) (a)efc ^ ee.

fc=i

< If f £ Cr(K, E), then f £ C(K, E(e)) for some e £ E +. According to the Kakutani-Krein Theorem, E(e) is linearly isometric and lattice isomorphic to C(Q) for some compact Hausdorff space Q. Therefore one can assume that f £ C(K, C(Q)). However, the spaces C(K, C(Q)) and C(K x Q) are isomorphic as Banach lattices. It remains to note that, according to the Stone-Weierstrass Theorem, the subspace of the functions

n

(a, q) ^ ^ (a)ek (q), where £ C (K) and ei,...,en £ C (Q), is dense in

k=i

C(K x Q). >

Definition 4. Let K be a compact Hausdorff space without isolated points and E be a relatively uniformly complete vector lattice. We denote by CDo(K, E) the set of E-valued functions on K each of which is the sum of two E-valued functions f and d with f £ Cr (K, E) and d £ co(K, E).

For a finite subset S of K and e £ E, xs ® e is in CDq (K, E). It is easy to see that CDr (K, E) is an ordered vector space under the pointwise order.

Lemma 5. Let K be a compact Hausdorff space without isolated points and E be a relatively uniformly complete vector lattice. Then, Cr (K, E) n Co(K, E) = {0}.

< Suppose the contrary; let 0 = f £ Cr (K, E) n Co(K, E). Assume that f (x) = 0. So there exists e £ E+ such that f £ C(K, E(e)). Then there exists a neighborhood V of x such that for y £ V we have ||f (y)||e > ||f (x)||e/2. But since x is not isolated, V is uncountable, which is a contradiction since f £ Co(K, E). >

It now follows that the decomposition of an element of CDQ (K, E)-space into a sum of an r-continuous function and one with countable support is unique. So CDr (K, E) deserves to be an Abramovich-Wickstead type space.

Lemma 6. Let K be a compact Hausdorff space without isolated points and E be a relatively uniformly complete vector lattice. Let p £ CDQ (K, E). Then p+ = sup(p, 0) exists in CD (K, E).

< Let p £ CD(K,E). Let r(k) = f+ (k) + [-f-(k) + h(k)] V (-f+ (k)) for each k £ K where f and h are continuous and discrete parts of p, respectively. By definitions f and h take their values in E(e) for some e £ E + . Let s(k) = (-f-(k) + h(k)) V (-f +(k)). Let e > 0 be given. Then there exists no £ N such that

{k £ K : e < ||s(k)||e} C {k £ K : — <

no

Indeed, if this were not true, then for some sequence (kn) in K, we would have e ^ ||s(kn)||e while ||h(kn)||e < n for all n £ N. By compactness of K, we can find a subnet (ka) of kn that converges to some ko £ K. As ||h(ka)||e ^ 0 in E(e), we have that

e < ||s(ka)|e = ||(-f— (ka) + h(ka)) V (-f + (k«))||e ^ || - f-(ko) V (-f +(ko))||e = 0 which is a contradiction. Hence r £ CDr (K, E) whenever p £ CDr (K, E). On the other hand,

r(k) = f +(k) + [-f-(k) + h(k)] V (-f + (k)) = [f + (k) - f-(k) + h(k)] V 0 = (p(k))+

for each k £ K .So r is indeed p+. Continuous part of r is f+, where f+(k) = (f (k))+ by uniqueness of decomposition. >

We summarize what we have from the previous proposition as follows:

Proposition 7. Let K be a compact Hausdorff space without isolated points and E be a relatively uniforly complete vector lattice. Then CDQ (K, E) is a vector lattice under the pointwise ordering: 0 ^ p £ CDQ (K, E) ^ 0 ^ p(k) in E for all k £ K.

Just like real-valued function space CDo(K) in [2], suprema and infima are easy to identify in CDr (K, E). We shall write h7 | h if the net h7 is increasing and sup(h7) = h.

Proposition 8. Let K be a compact Hausdorff space without isolated points and E be a relatively uniformly complete vector lattice. If h7 | h in CDr (K, E), then h7 (k) | h(k) in E for all k £ K.

< Let ko be an arbitrary but fixed point of K. Then h(ko) is an upper bound of {h7(ko) : Y £ r} in E(e) for some e £ E+. Let v be another upper bound for {h7(ko) : Y £ r}. If v A h(ko) = h(ko), then the proof is obvious. On the other hand, if v A h(ko) < h(ko), then we can find some 0 < ei £ E(e) such that v A h(ko) + ei ^ h(ko). Then h - %fc0 ^ ei is an upper bound in CD^(K, E) for the family {h7 : Y £ r}, contradicting the definition of h. >

From the proposition above, we conclude that order convergence in CDq (K, E) is pointwise, order convergence in E.

3. Linear operators on Cr (K, E) and co(K, E)

Throughout this section, unless stated otherwise, E will denote a relatively uniformly complete vector lattice and for a vector valued function f, Xk ® f will denote the function which takes f (k) at k and 0 otherwise. In this section we give two characterizations about the regular and order continuous linear operators from CD£j (K, E) into a Dedekind complete vector lattice F.

We start with the following lemma which will be used in the sequel.

Lemma 9. Let K be a compact space and F be a Dedekind complete vector lattice. Then for every regular operator T : Cr (K, E) ^ F there exists a regular operator T' : C(K) ^ Lr (E, F) such that

T(< <8) e) = T'(<)e for all < £ C(K) and e £ E.

The correspondence T ^ T' is linear positive, and one-to-one.

< It is sufficient to consider positive linear operators. Let T : Cr(K, E) ^ F be a positive linear operator. For each < £ C(K) and e £ E, the function <® e defined by <® e(k) = <(k)e belongs to Cr (K, E). We put

T(< <8) e) = T'(<)e for all < £ C(K) and e £ E.

For fixed < £ C(K), the mapping T'(<) : e ^ T'(<)e of E into F is evidently linear. Moreover, if 0 ^ e £ E and 0 ^ < £ C(K), then T'(<)e = T(< ® e) ^ 0, therefore T'(<) £ L+(E, F). Thus, the mapping T' : < ^ T'(<) of C(K) into Lr (E, F) is linear and positive.

It is easy to verify that the mapping T ^ T' is linear and positive. In order to prove that this mapping is one-to-one, let S : Cr (K, E) ^ F be a positive linear operator such that

S(< ® e) = T'(<)e, for < £ C(K) and e £ E.

Let f £ Cr (K, E). Then by Theorem 3, there exists a sequence (fn) of the form ^ <<X)e» (finite sum) with £ C(K) and e» £ E converging relatively uniformly to f. Then T(fn) = S(fn) for every n. On the other hand T and S are relatively uniformly continuous on Cr (K, E), therefore

T(f) = lim T(fn) = lim S(fn) = S(f),

n—n—

consequently T = S. >

Theorem 10. Let K be a compact topological space and F be a Dedekind complete vector lattice. Then there exists a lattice isomorphism T' ^ ^ between the set of regular operators T' : C(K) ^ Lr (E, F) and the set of countably additive quasiregular Borel measures ^ : K ^ Lr (E, F) given by the equality

T'(f) = J fd^, for every f £ C(K).

< Proof directly follows from Theorem 1, since Lr (E, F) is a Dedekind complete vector lattice. >

Let F be another Dedekind complete vector lattice and ^ £ qca(K, Lr(E, F)). Then the integral : C(K) ^ E can be extended to Cr (K, E). We can view the algebraic tensor

product C(K) < E as the subspace in Cr (K, E), consisting of the mappings k ^ Sn=i pi(k)ej (k G K) where ei G E and p G C(K). Define on C(K) < E by the formula

(n \ n .

Pi < eA := ^ eW p d^. i=i / i=i K

K

If / G Cr (K, E), then by Theorem 3 there exist e G E + and a sequence (/n) C C(K) < E such that

sup |f (k) - fn(k)| < 1 e. fcSK n

Put by definition

J /d^:= IM(f) := o-lim I/). K

It can be easily seen that this definition is correct even for an arbitrary order bounded finitely additive vector measure

Theorem 11. For any linear operator T G Lr (Cr (K, E), F) there exists a unique vector measure ^ := ^y G qca(K, Lr (E, F)) such that

T/ = 1 /(k) d^(k) (/ G Cr(K, E)).

K

The correspondence T ^ ^y is a lattice isomorphism of Lr (Cr (K, E), F) onto qca(K, Lr(E, F)).

< See [6, Theorem 2.1.14(5)]. >

Theorem 12. There exists a lattice isomorphism T ^ T' between the space of regular operators T : Cr (K, E) ^ F and the space of regular operators T' : C (K) ^ Lr (E, F) given by the equality

T(p < e) = T'(p)e, for p G C(K) and e G E.

If T and T' are in correspondence, then there exists a unique common countably additive quasiregular Borel measure ^ := ^y : K ^ Lr (E, F) such that

and

T(/) = y /d^, for / G Cr (K, E), T'(p) = J pd^, for p G C(K).

In particulal, the correspondence T ^ ^y is a lattice isomorphism of Lr(Cr(K, E),F) onto qca(K, Lr(E, F)).

< Let first T : Cr (K, E) ^ F be a regular operator. Let T' : C(K) ^ Lr (E, F) be the regular operator corresponding to T (Lemma 9) by the equality

T(p < x) = T'(p)x, for p G C(K) and x G E.

We know that the correspondence T ^ T' is linear, positive, and one-to-one. We have

T(p < x) = T'(p)x = ^ p d^ x

for every x € E, therefore

T(ф ® x) = J ф <X> x d^, for every ф € C(K) and x € E.

Conversely, let T' : C(K) ^ Lr (E, F) be a regular operator, and let ^ : K ^ Lr (E, F) be the countably additive quasiregular measure corresponding to T' by Theorem 10. If we put

T(/ ) = У /ф, / € Cr(K,E), then T : Cr (K, E) ^ F is a regular operator and we have

T(ф <8) x) = T'(^)x, for ф € C(K) and x € E.

It remains to apply Theorem 11. >

Now we give the following definition which will be useful in the sequel.

Definition 13. Let K be a non-empty set and F be a Dedekind complete vector lattice. Then we define

(1) c0(N, E) = |(e„) С E : 3e € E+ such that e„ € E(e)Vn and ||e„||e ^ 0},

те

(2) 1i[K,Lr(E,F)] is the space of maps а : K ^ Lr(E, F) such that £ |a(k„)|(|en|)

n=1

exists in F for all sequences (kn) € K and (en) € c0 (N, E).

те m

As usual, ^ |a(kn)| (|en|) is the supremum of the sums ^ |a(kn)|(|en|). Clearly,

n=1 n=1

1i[K, Lr (E, F)] is a vector lattice under the pointwise operations and ordering.

Theorem 14. Let K and F be as above. Then Lr(co(K, E),F) is lattice isomorphic to 1i[K,Lr (E,F)].

< Let ф : Lr(c0(K,E),F) ^ 11[K,L(E,F)] be defined by 0(G)(k)(e) = G(Xk ® e) for each G € Lr(co(K, E), F), k € K and e € E. Then ф(С)(к) is a regular operator from E into F as ф(в+ )(k) and ф(в-)(к) are positive for each regular operator G. Thus ф(в) is a map from K into Lr (E, F) and ф^)(к)^) ^ 0 whenever e ^ 0 and G ^ 0, i. e. ф^)(к) is positive for all G ^ 0.

те

Let us recall that ф^) should also satisfy ^ ^(G)(kn)|(|en|) € F for all sequences

n=1

(kn) € K and (en) € c0(N,E). Let G € Lr(c0(K,E),F). Then we have

m m / m \ /те \

E IФ(G)(kn)|(|en|) = E |G|(Xfcn®|e„|) = |G| EXfc„ ® Ы < |G| EXfc„ <8 |e„| € F,

n=1 n=1 \ra=1 / \ra=1 /

therefore

те m

E ^(G)(k„)|(|e„|) = supE |G|(Xfen <8 |en|) € F.

m

n=1 n=1

Thus the map ф^) we have defined belongs to l1[K, L(E,F)].

It is easy to verify that ф is a linear mapping. We now show that it is bipositive. Suppose that ф^) ^ 0 for some G € Lr(c0(K, E), F), and take 0 ^ / € c0(K, E). As £ Xk ® / /

in co(K, E), we have £ G(Xfc ® /) ^ G(/). By definition G(Xfc ® /) = ф^)^)/(k)) ^ 0

and thus G(/) ^ 0 for each 0 ^ / € co(K, E), i. e. G ^ 0.

Let now 0(G) = 0 for some G G Lr(c0(K, E), F). Then G(xk < /) = 0 for each k G K and

and 0 < / G Co(K, E). As £ Xfc < / Ts / in Co(K, E), we have 0 = £ G(Xfc < /) ^ G(/) or

fees fees

G(/) = 0. The fact that Co(K, E) is vector lattice leads to G = 0.

To show that 0 is surjective, let 0 ^ a G 1i[K, Lr (E, F)]. Let / G Co(K, E). Then there

exists an at most countable subset (kn) of K such that /(k) = 0 for all k = kn and there

exists some e G E + such that /(kn) G E(e) for each n and ||/(kn)||e ^ 0. Hence we can define

G(/) = £ a(kn)(/(kn)), neN

which certainly belongs to F as /(kn) G c0(N, E). We now verify that 0(G) = a. Let 0 ^ e G E, then

0(G)(ko)(e) = G(xfco < e) = £ a(ko)(xfco < e) = a(ko)(e).

neN

Since e G E is arbitrary, we conclude that 0(G)(ko) = a(ko) and ko is arbitrary, we have 0(G) = a. Since 1i[K, Lr (E, F)] is a vector lattice, the proof of surjectivity of 0 is now complete. >

4. Main resalts

Now we are in a position to give one of the first main result of this note as follows:

Theorem 15. Let K be a compact Hausdorff space without isolated points and F be a Dedekind complete vector lattice. Then Lr(CDq(K, E),F) is lattice isomorphic to qca(K, Lr (E, F) © li [K, Lr (E, F)] with the dual order on this direct sum defined by

a) ^ 0 ^ ^ ^ 0 and a ^ 0 and ^({k}) ^ a(k)

for all k G K, which if we identify a with a discrete measure on K, is precisely requiring that p ^ a ^ 0.

< Let T G Lr(CDr (K, E),F). Then certainly T splits into two regular operators Ti and T2, where Ti : Cr (K, E) ^ F and T2 : Co(K, E) ^ F. By Theorem 12 there exists a measure ^ G qca(K, Lr (E, F) such that Ti can be identified with On the other hand, by Theorem 14 there exists a map a G li [K, Lr (E, F)] such that T2 can be identified with a. We thus have a map from Lr(CDg(K, E),F) into qca(K, Lr(E, F) © 1i[K,Lr(E, F)].

Now suppose that ^ G qca(K, Lr (E, F)) and a G ¿i[K, Lr (E, F)]. We can certainly define an operator p by

p(/) = / /i d^ + £ a(kn)(/2(kn)),

neN

for / = /i + /2 G Cr (K, E) © co(K, E). The map we have defined from qca(K, Lr (E, F)) © 1i[K, Lr (E, F)] into Lr (CDq (K, E), F) is easily seen to be lattice isomorphism by Theorem 12 and Theorem 14. >

Now we give the following definition which will be used in our final result. Definition 16. Let K be a compact space and F be a Dedekind complete vector lattice. Then we set 1i(K, Ln(E, F)) the set of all maps ^ from K into Li(E, F) satisfying

(1) sup |^(k)|(|(/(k)|) exist in F for all e G E + and / G C(K, E(e)) © co(K,E(e)),

11/lle^i keK

(2) £ |^(k)|(/a(k)) 1« 0 whenever /a j 0. fceK

As usual, (k)|(|(/(k)|) is the supremum of the sums ^ |^(k)|(|/(k)|) where S is a

keK kes

finite subset of K. Evidently, l1(K, L^(E, F)) is a vector lattice under pointwise operations.

We close this section with a result of this note about order continuous operators on CD£j (K, E)-spaces.

Theorem 17. Let K be a compact Hausdorff space without isolated points and F be a Dedekind complete vector lattice. Then L^CD^(K, E),F) is lattice isomorphic to l1(K,L;(E,F)).

< Define 0 : Ln(CDg(K, E),F) ^ l1(K,Ln(E,F)) via 0(G)(k)(e) = G(Xk ® e) for each G G Ln(CD5(K,E),F), k G K and e G E. Then 0(G)(k) is order bounded, since 0(G+)(k) and 0(G-)(k) are order bounded F-valued operators for each G on CDq(K, E). If ea j 0 in E, then xk ® ea j 0 in CD£(K, E) for each k G K. This gives that 0(G)(k)(ea) = G(xk ® ea) is order convergent to 0 so that 0(G)(k) G L;(E,F) for each G G L^CDg(K, E),F). Thus 0(G) is a map from K into L;(E, F) and 0(G)(k)(e) ^ 0 whenever e ^ 0 and G ^ 0, i. e., 0(G)(k) is positive for all G ^ 0.

Now we will show that 0(G) is an element of l1 (K, L;(E, F)). Let S be a finite subset of K and G G L^CDJ (K, E), F). Then

E |0(G)(k)|(|/(k)|) = E |0(G+ - G-)(k)|(|/(k)|)

kes kes

< E0(G+)(k)(|/(k)|) + E0(G-)(k)(|/(k)|)

kes fces

= EG+(Xk ®|/1) + E G-(Xk ®|/1) = G+(EXk ®|/1) + G- (EXk ®|/1)

kes kes Vkes / Vkes /

for each / G CD0 (K, E). Hence we get

E |0(G)(k)|(|/(k)|) < G+(|/1) + G-(|/1) = |G|(|/1) kes

as X] Xk ® |/1 Ts |/1, G+ and G- are order continuous.

kes

Let e be an arbitrary but fixed element of E +. Then

sup E |0(G)(k)|(|/(k)|) < sup |G|(|/1) < |G|(e) G F,

11/lie <1 k 11/lie<1

as |/1 < ||/||ee.

So far we have shown that 0(G) satisfies the first condition of Definition 16. Also we have to show that

E |0(G)(k)|(/a(k)) 1« 0 k

for each /a G CDq (K, E) such that /a j 0. It is sufficient to verity the claim for a positive

G G L;(CDS(K, E), F). Let 0 ^ G G L^CDJ(K, E),F) and /Q j 0 in CD£(K, E). For a

fixed a, we have ^ Xk ® /a Ts /«. Since G is order continuous and positive, we have kes

G E Xk ® /a = E G(Xk ® /a) T G(/a).

Vkes / kes

Thus

£ |0(G)(k)|(/a(k)) = £ 0(G)(k)(/a(k)) = £ G(Xk ® /a) = G/a) j 0.

keK keK keK

Hence the map 0(G) we have defined belongs to 1i(K, Ln(E, F)).

It is easy to see that 0 is linear. We now show that it is bipositive. Certainly 0(G) ^ 0 whenever G ^ 0. Now assume that 0(G) ^ 0 for some G G Ln(CDo (K, E),F) and take

0 < / G CDS(K, E). As £ xfc ® / Ts / in CD£(K, E), we have £ G(xk <8 /) ^ G(/). By

kes kes

definition, G(xk <8) /) = 0(G)(k)(/) ^ 0 and thus G(/) ^ 0 for each 0 < / G CD^j(K, E), i. e., G ^ 0. We now show that 0 is one-to-one. Let 0(G) = 0 for some G G L^CD^(K,E),F). Then G(xk ® /) = 0 for each k G K and 0 ^ / G CDo(K, E). As G is order continuous and

£ Xk ® / Ts /, this gives that 0 = £ G(xk ® /) ^ G(/) or G(/) = 0. As CDo(K, E) is a kes kes

vector lattice, we get G = 0.

To show that 0 is surjective, take an arbitrary 0 ^ a G li(K, Ln(E, F)) and define

G : CDS(K,E)+ ^ F by G(/) = £ a(k)(/(k)). Then G is additive on CDo(K, E) and

keK

G(/) = G(/+) - G(/-) extends G to CDg(K, E). We now verify that 0(G) = a. If 0 < e G E, then

0(G)(ko)(e) = G(xko ® e) = £ a(k)(xko ® e)(k) = a(ko)e.

keK

Since e G E is arbitrary, we conclude that 0(G)(ko) = a(ko) and ko is arbitrary, we have 0(G) = a. >

Acknowledgement

I would like to thank Prof. A. G. Kusraev as he read this manuscript and made many valuable suggestions and comments on it.

References

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3. Ercan Z., Alpay S. CDo(K, E) and CDW (K, E)-spaces as Banach lattices // Positivity.—2000.—V. 4.— P. 213-225.

4. Ercan Z., Onal S. Kakutani-Krein compact space of CDw (X )-spaces interms of X ®{0, 1} // J. of Math. Anal. and Appl.—2006.—V. 313, № 2.—P. 611-631.

5. Aliprantis C. D., Burkinshaw O. Positive Operators.--New York: Acad. press, 1985.—xvi, 367 p.

6. Kusraev A. G., Kutateladze S. S. Subdifferentials: Theory and Applications.—Dordrecht: Kluwer Academic Publishers, 1995.—408 p.

7. Wright J. D. M. An algebraic characterization of vector lattices with the Borel regularity property // J. London Math. Soc.—1973.—V. 7, № 2.—P. 277-285.

Received April 2, 2008.

Faruk Polat Department of Mathematics, Middle East Technical University Ankara, 06531, TURKEY E-mail: [email protected]