On Multipliers andL- andM-Projections in Banach Lattices and Köthe Function Spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

206, 83]102 Ž1997.

AY975197

On Multipliers and L- and M-Projections in Banach Lattices and Kothe ¨ Function Spaces Lech Drewnowski* Faculty of Mathematics and Computer Science, A. Mickiewicz Uni¨ ersity, Poznan, ´ Poland Institute of Mathematics

and Anna Kaminska ´ † and Pei-Kee Lin‡ Department of Mathematics Sciences, The Uni¨ ersity of Memphis, Memphis, Tennessee 38152 Submitted by Richard M. Aron Received December 13, 1995

INTRODUCTION For a Banach space E, let L Ž E . be the Banach algebra of Žbounded linear. operators on E, and let Ext E and Ext E* be the sets of extreme points of the closed unit balls in E and in its dual E*. An operator T g L Ž E . is called a multiplier if every p g Ext E* is an eigenvector of the adjoint T *, that is, T *Ž p . s aT Ž p . p for some number aT Ž p .. Let MultŽ E . stand for the set of multipliers on E. A projection P g L Ž E . is said to be an M-projection Žresp., L-projection. if 5 x 5 s max5 P Ž x .5, 5 x y P Ž x .54 Žresp., 5 x 5 s 5 P Ž x .5 q 5 x y P Ž x .5. for every x g E wBe1, Be2x. Let I denote the identity operator on E. We say that a multiplier T Žresp., a projection P . on E is trivial if T s cI for some scalar c Žresp., P s I or P s 0.. Given a vector lattice E, a sublattice F of E is called an ideal if it contains every x g E with < x < F y for some y g F. If x g E, then I Ž x . [ * E-mail address: [email protected]. † E-mail address: [email protected]. ‡ E-mail address: [email protected]. 83 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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 y g E: < y < F n < x < for some n g N4 is the ideal generated by x. An ideal F of E is said to be a band if whenever A ; F and sup A s x exists in E, then x g F. Two elements x, y g E are said to be disjoint, x H y, if < x < n < y < s 0. If B / A ; E, then AH [  x g E: x H y, ; y g A4 is a band in E. A band F of E is called a projection band if E s F q F H . The associated projection P: E ª F with kernel F H is then called a band projection. It is well known that every band in E is a projection band if E is order complete, and if E is a Banach lattice and P is a band projection on E, then P* is a band projection on E* Žcf. wA-Bx.. Moreover, if E is order s-complete, then for every x g E the Žprinciple. band generated by x is a projection band. The associated band projection will be denoted by Px , and the complementary band projection I y Px will be often denoted by Q x . Further, for x g E, we let C Ž x . stand for the set of all Žprojection. components of x, i.e., elements of the form PuŽ x ., where u g X Žit suffices to consider only 0 F u F < x <.. Let Ž V, S, m . be a s-finite measure space. By L0 Ž m . s L0 Ž V, S, m . denote a vector lattice of all m-measurable complex or real functions on V. A Kothe ¨ function space E over Ž V, S, m . is the Banach lattice which, as a vector lattice, is an ideal of vector lattice L0 Ž m .. Without loss of generality, let’s further assume that supp E s V Žcf. wK-A, L-Tx.. Any Kothe ¨ space E is order complete, hence every band in E is a projection band. Also, it is well known that the band projections on E coincide with the characteristic projections PA : f ª f x A , where A g S and x A denotes the characteristic function of A. A Kothe ¨ function space E is said to have the Fatou property if whenever Ž f n . is a norm bounded sequence in E such that 0 F f n ­ f g L0 Ž m ., then f g E and 5 f 5 s lim n 5 f n 5. E has order semicontinuous norm if 5 f n 5 ª 5 f 5 whenever 0 F f n ­ f g E. An element f g E is said to have an absolutely continuous norm if for any sequence f n ª 0 and < f n < F < f < we have 5 f n 5 ª 0. The norm in E is called order continuous if every element in E has absolutely continuous norm Žcf. wK-A, L-Tx.. The Kothe dual of E is the ¨ Kothe space E9 [  g g L0 Ž m . : HV < fg < d m - ` for all f g E4 , equipped ¨ with the norm defined by 5 g 5 s sup HV < fg < d m : f g E, 5 f 5 F 14 . A Kothe ¨ function space E is said to be rearrangement in¨ ariant Žr.i.. if whenever f g E, g g L0 Ž m ., and f and g are equimeasurable, then g g E and 5 f 5 s 5 g 5. Recall that f and g are called equimeasurable if < f < and < g < have identical distributions; that is, m t g V: < f Ž t .< ) l4 s m t g V: < g Ž t .< ) l4 for all l G 0 Žcf. wL-Tx.. The following results on L- and M-projections in Banach spaces are very well known facts Žsee wBe1, Be2, and HWWx.. Ž1. If P is an M-projection Žresp., L-projection., then P* is an L-projection Žresp., M-projection., and conversely wBe1, Proposition 1.5x.

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85

Ž2. Every M-projection is a multiplier. Ž3. No complex Banach space may admit at the same time a nontrivial M-projection and a nontrivial L-projection. The same holds true for all real Banach spaces except those linearly isometric to l`2 wBe1, Theorem 1.13x. Ž4. Lebesgue spaces L p Ž m . for 1 - p - ` have no nontrivial M- or L-projections wBe1, Theorem 6.2x. The L-projections on L1Ž m ., and the M-projections on L`Ž m ., coincide with the band projections, i.e., with the characteristic projections PA , where A g S Žcf. wBe2, Proposition 4.9, Be1, Example 2, p. 10x.. Furthermore, the multipliers on L`Ž m . coincide with the multiplication operators Mh : f ª h ? f, where h g L`Ž m . Žcf. wBe1, Example 2, p. 55x.. In this article, we study multipliers and M- and L-projections on Žreal or complex. Banach lattices and, in particular, Kothe function spaces. In¨ spired by the results mentioned in Ž4., we consider the following questions. QUESTION 1. Let E be a Banach lattice. Is e¨ ery M-projection Ž resp., L-projection. on E a band projection? QUESTION 2. Let E be a Kothe ¨ space of m-measurable functions. Is then e¨ ery multiplier on E a multiplication operator induced by a bounded m-measurable function? QUESTION 3. Let E be a rearrangement in¨ ariant Kothe ¨ space of m-measurable functions. Suppose there is a nontri¨ ial L-projection Ž resp. M-projection. on E. Is then E s L1Ž m . Ž resp. E s L`Ž m ..? The paper consists of three sections. In Section 1, we deal with complex Banach lattices and get some results crucial for further studies of multipliers and M- and L-projections in complex spaces. We show that M- and L-projections on E are band projections. We also show that the same holds for L-projections on real Banach lattices with strictly monotone norm. In Section 2, we work with Žreal or complex. Kothe ¨ function spaces E. We prove that the multipliers on E which are multiplication operators are exactly the operators in the closed linear span of the characteristic M-projections. We also show that E coincides with L1Ž m . or L`Ž m ., whenever E is rearrangement invariant and admits a nontrivial characteristic L- or M-projection, respectively. These two results combined with the results of the first section yield the affirmative answer to all three questions raised above in the case of complex spaces. In the examples presented further in Section 2, it is shown that the answers to Questions 1 and 2 are ‘‘no,’’ in general for real spaces E. However, under certain restrictions on the Kothe ¨ function space E Žreal or complex., it is proved that every multiplier on E is a multiplication

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operator. A number of consequences is derived from this result. For instance, if E is rearrangement invariant and has order semicontinuous norm, then it has no nontrivial multipliers. In Section 3, we consider Orlicz spaces. From the results of Section 2 we first deduce that every multiplier on an Orlicz space is trivial. We then proceed to proving that the only Orlicz space with nontrivial L-projections is L1Ž m .. Let’s now agree that in the sequel we will omit the adjectives complex or real when dealing with spaces of either type.

1. COMPLEX BANACH LATTICES We will first prove the following auxiliary lemma. Although it is known in the real case Že.g., wL-Z, Theorem 40.2, La, Theorem 1.1.8; or Lip, Theorem 2x., its extension to complex lattices is not automatic. LEMMA 1.1. Let E be an order s-complete ¨ ector lattice, and let x g E. Then for e¨ ery y g I Ž x . and e¨ ery number « ) 0 there exists z g lin C Ž x . such that < y y z < F « < x <. In consequence, if E is a normed ¨ ector lattice, then lin C Ž x . is norm dense in I Ž x .. Proof. Let k g N be such that < y < F k < x < and take any number 0 h - 1. By what was said above, we can find elements w and ¨ in lin C Ž< x <. of the form n

ws

Ý a j Pu Ž < x <. j

n

and

js1

¨s

Ý bj Pu Ž < x <. , j

js1

where 0 F u j F < x < are pairwise disjoint, Pu jŽ< x <. ) 0, and a j and bj are scalars, such that < y y w< F h< x<

and

< x y ¨ < F h < x <.

From the first estimate it follows that n

Ý < a j < Pu Ž < x <. s < w < F < y < q h < x < F Ž k q 1. < x < , j

js1

while the second estimate yields that for every j,

Ž 1 y < bj < . Pu Ž < x <. s j

Pu jŽ x . y bj Pu jŽ < x < . F Pu jŽ x . y bj Pu jŽ < x < .

s Pu jŽ < x y ¨ < . F h Pu jŽ < x < . ,

87

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hence < bj < G 1 y h. Denoting c j s 1rbj , we have h h c j Pu jŽ x . y Pu jŽ < x < . F ? Pu jŽ < x < . F ? P Ž < x <. . < bj < 1 y h uj Now, define n

zs

Ý a j c j Pu Ž x . . j

js1

Then < z y w< s

n

Ý a j Ž c j Pu Ž x . y Pu Ž < x <. . j

j

js1

F

h 1yh

h

n

?

Ý < a j < Pu Ž < x <. F Ž k q 1. 1 y h ? < x < . j

js1

Therefore, < y y z < F < y y w < q < w y z < F h < x < q Ž k q 1.

h 1yh

? < x <,

which is F « < x < if h is chosen sufficiently small. We will also need the following lemma. LEMMA 1.2. Let F be a complex Banach space, and let T be an operator on F such that e¨ ery point in Ext F is an eigen¨ ector of T. Moreo¨ er, let F s X [ Y be a direct sum decomposition of F such that 5 x q cy 5 s 5 x q y 5

for all x g X , y g Y , and c g C with < c < s 1.

If w g Ext F and w s u q ¨ , where u g X and ¨ g Y, then there is a g C such that T Ž z . s az for all z g lin  u, ¨ 4 . Proof. By the assumption on F, if < c < s 1, then the map x q y ª x q cy Ž x g X, y g Y . is a surjective linear isometry in F. In consequence, u q c¨ g Ext F; hence, by the assumption on T, T Ž u q c¨ . s a c Ž u q c¨ . for some constant a c . Clearly, we may assume that u / 0 and ¨ / 0. Then u and ¨ , as well as x s u q ¨ and y s u y ¨ are a basis of lin u, ¨ 4 , and to prove the assertion it is enough to show that a1 s ay1. Now, if < c < s 1, then 2T Ž u q c¨ . s T Ž Ž 1 q c . x q Ž 1 y c . y . s a1 Ž 1 q c . x q ay1 Ž 1 y c . y, and 2 a c Ž u q c¨ . s a c Ž 1 q c . x q a c Ž 1 y c . y. From this, taking any c / "1 with < c < s 1, we obtain a c s a1 s ay1.

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We are now ready to prove our first result on multipliers. THEOREM 1.3. Let T be a multiplier of a complex Banach lattice E. Then for e¨ ery p g Ext E*, T * Ž q . s aT Ž p . q

for all q g I Ž p . .

Ž ).

In consequence, aT Ž p1 . s aT Ž p 2 .

whene¨ er p1 , p 2 g Ext E* and < p1 < n < p 2 < / 0.

Proof. From Lemma 1.2 it follows directly that if Q is a band projection on E*, then T * Ž q . s aT Ž p . q

for all q g lin  Q Ž p . , Ž I y Q . Ž p . 4 .

Hence the equality in Ž). holds for all q g C Ž p ., and also for all q g lin C Ž p .. Since T * is continuous and, by Lemma 1.1, lin C Ž p . is norm dense in I Ž p ., the equality in Ž). holds for all q g I Ž p .. COROLLARY 1.4. Let T be a multiplier on an order s-complete complex Banach lattice E. Then Px (T Ž x . s T Ž x . for e¨ ery x g E. Hence if x, y g E and x H y, then T Ž x . H T Ž y .. Proof. Let x g E and denote Q s I y Px . If QT Ž x . / 0 then, by the Krein]Milman theorem, there exists p g Ext E* such that ² QT Ž x ., p : / 0. By Theorem 1.3, T *Q*Ž p . s aT Ž p . Q*Ž p .. Therefore,

² QT Ž x . , p: s² x, T *Q* Ž p .: saT Ž p .² x, Q* Ž p .: saT Ž p .² Q Ž x . , p: s0; a contradiction. The next result provides a positive answer to Question 1 in the case of complex lattices. THEOREM 1.5. E¨ ery M-projection and e¨ ery L-projection on a complex Banach lattice E is a band projection. Proof. Ža. Let P be an M-projection on E. Assume first that E is order s-complete. Since P* is an L-projection, so P*Ž p . s 0 or p for every p g Ext E*. Hence P is a multiplier with a P Ž p . s 0 or 1 for every p g Ext E*. In order to prove the assertion it is enough to show that x H PŽ y.

whenever x, y g E and P Ž x . s 0.

Suppose < x < n < P Ž y .< ) 0 for some x and y with P Ž x . s 0. Let x s x 1 y x 2 q ix 3 y ix 4 and P Ž y . s z1 y z 2 q iz 3 y iz 4 , where all x j , z j G 0,

MULTIPLIERS AND PROJECTIONS

89

and x 1 H x 2 , x 3 H x 4 , z1 H z 2 , and z 3 H z 4 . Replacing x by yx or "ix andror y by yy or "iy if necessary, we may assume that z1 n x 1 s z ) 0. Hence there exists p g Ext E* with ² z, p : / 0. Thus for q s < p < we have ² z, q : ) 0. Note that if a functional r g E* is positive and w g E, then Re² w, r : s ² u, r :, where u is the real part of w in E. Since q is positive, both PzU1 Ž q . and QUz 1Ž q . are positive. Therefore, Re ² Py, Ž PzU1 y QUz 1 . Ž q .: s² z1 y z 2 , Ž PzU1 y QUz 1 . Ž q .: s² Pz 1Ž z1 . , q: q² Q z 1Ž z 2 . , q: s ² z1 q z 2 , q : G ² z, q : ) 0. Hence P U (Ž PzU1 y QUz 1 .Ž q . / 0. By Theorem 1.3, a P Ž p . s 1 and P U ( Ž PxU y QUx .Ž q . s Ž PxU y QUx .Ž q .. Thus, since Px s 0, we obtain 1 1 1 1 0 s² Px, Ž PxU1 y QUx 1 . Ž q .: s² x, Ž PxU1 y QUx 1 . Ž q .: s Re ² x, Ž PxU1 y QUx 1 . Ž q .: s ² x 1 q x 2 , q : G ² z, q : ) 0, a contradiction. Now, consider the case of a general complex Banach lattice E. Then P** is an M-projection on E**, and E** is an order complete complex Banach lattice. By the previous part of the proof, P** is a band projection, hence so is P s P** < E . Žb. Let P be an L-projection on E. Then P* is an M-projection, hence, by the previous case, P* is a band projection. So also P s P** < E is a band projection. A Banach lattice E is said to have strictly monotone norm if 5 x 5 - 5 y 5 whenever < x < - < y < Žcf. wB-Hx.. THEOREM 1.6. E¨ ery L-projection on a Ž real or complex . Banach lattice E with strictly monotone norm is a band projection. Proof. Since the complex case is already covered by Theorem 1.5, we give the argument only for the real case. Let P be an L-projection on E and denote Q s I y P. It is enough to show that < Px < n < Qx < s 0 for every x g E. Suppose it is not so for some x. Without loss of generality, we may assume that Ž Px .qnŽ Qx .qs y ) 0. Then 5 Px 5 q 5 Qx 5 s 5 Px y Qx 5 F 5 Px y y 5 q 5 Qx y y 5 F 5 Px 5 q 5 Qx 5 . Hence 5 Px y y 5 s 5 Px 5, which contradicts the strict monotonicity of the norm in E.

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¨ 2. KOTHE FUNCTION SPACES We start with two results on multiplication operators on Kothe ¨ function spaces. PROPOSITION 2.1.

Let E be a Kothe ¨ function space.

Ža. Let h g L0 Ž m .. Then hf g E for e¨ ery f g E if and only if h g L`Ž m .; in this case the multiplication operator Mh : E ª E; f ª hf is continuous and 5 Mh 5 s 5 h 5 ` . Žb. The map M: h ª Mh is an isometric algebra isomorphism from L`Ž m . onto the subalgebra lin PA : A g S4 of L Ž E .. Proof. Since Žb. is a straightforward consequence of Ža., we only verify Ža.. Let h / 0. If Mh maps E into itself, then Mh is continuous, as follows from the closed graph theorem and the fact that the norm convergence in Kothe ¨ function spaces implies the convergence in measure on every set of finite measure wK-A, IV, Sect. 3, Theorem 1x. Take any 0 - c - ess sup < h <. There exists a non-null set A g S such that c x A F < h < and x A g E. Hence 5 Mh 5 5 x A 5 G Mh Ž x A . s 5 h x A 5 G c 5 x A 5 and so 5 Mh 5 G c. In consequence, h g L`Ž m . and 5 Mh 5 G 5 h 5 ` . Since the opposite inequality is obvious if h g L`Ž m ., the proof is complete. The following proposition was proved by Zaanen in wZax. PROPOSITION 2.2 wZa, Theorem 8x. For an operator T on a Kothe ¨ function space E, the following conditions are equi¨ alent. Ža. supp T Ž f . ; supp f for e¨ ery f g E. Žb. T Ž f x A . s T Ž f . x A for all f g E and A g S. Žc. T Ž fg . s T Ž f . g for all f g E and g g L`Ž m .. Žd. There exists h g L`Ž m . such that T s Mh . Given a Kothe ¨ function space E, define S M Ž E . s  A g S: PA is an M-projection on E 4 , S LŽ E . s  A g S: PA is an L-projection on E 4 . It is clear that S M Ž E . and S LŽ E . are subalgebras of S containing all m-null sets Žcf. wBe1, 1.9x.. Moreover, also the following holds Žcf. wBe1, Theorem 1.10x..

MULTIPLIERS AND PROJECTIONS

LEMMA 2.3.

91

Let E be a Kothe ¨ function space.

Ža. If Ž A n . is a sequence in S LŽ E . and A n ­ A, then A g S LŽ E . and PA nŽ f . ª PAŽ f . in norm for e¨ ery f g E. In particular, S LŽ E . is a sub-s-algebra of S. Žb. If E has order semicontinuous norm, then S M Ž E . is a sub-s-algebra of S. Proof. Since Žb. is rather straightforward, we only indicate how to prove Ža.: By wBe1, Theorem 1.10Žii.x, there is an L-projection P on E such that PA n ª P pointwise on E. It follows that, for every f g E, PA kŽ f . s PA kŽ Pf . for all k g N. Hence PAŽ f . s PAŽ Pf ., and we easily conclude that Pf s PAŽ f .. Recall that, for every Banach space E, the set MultŽ E . is a closed commutative subalgebra of the algebra L Ž E . wBe1, p. 54x. THEOREM 2.4. Let E be a Kothe ¨ function space with order semicontinuous norm. If T g MultŽ E . and T s Mh for some h g L`Ž m ., then h g L`Ž V, S M Ž E ., m .. Thus the map M: h ª Mh is an isometric algebra isomorphism from L`Ž V, S M Ž E ., m . onto the subalgebra MultŽ E . l lin PA : A g S4 of L Ž E ., consisting of all those multipliers which are multiplication operators. Proof. Let T g MultŽ E . and T s Mh for some h g L`Ž m .. We need to show that h is S M Ž E .-measurable. It is enough to see that A Ž b, « . [  t g V : h Ž t . y b F « 4 g S M Ž E . for every b g C and « ) 0. We first verify the following claim. Ž). If p g Ext E*, then ² f, p : s 0 for every f g E with supp f ;  t: < hŽ t . y aT Ž p .< ) « 4 for some « ) 0. Denote c s aT Ž p . and suppose an f g E is as required in Ž). but ² f, p : / 0; we may also assume that 5 f 5 s 1. Let 0 - h - « <² f, p :<Ž1 q <² f, p :<.y1 . Since h g L`Ž m ., using a compactness argument we can find bj g C with < bj y c < G « y h and pairwise disjoint A j g S so that < hŽ t . y bj < - h for a.a. t g A j , where j s 1, . . . , n, and A1 j ??? j A n s supp f. For each j choose s j g C with < s j < s 1 so that s j ²Ž bj y c . f x A j , p : s < bj y c < <² f x A , p :<. Then g [ Ý njs1 s j f x A g E and 5 g 5 s 5 f 5 s 1. Morej j over,

a[

¦

n

;

Ý Ž bj yc . g x A , p j

js1

n

s

Ý < bj yc < js1

² f x A , p : G Ž « yh . ² f , p : j

DREWNOWSKI, KAMINSKA ´ , AND LIN

92 and

b[

¦

n

;

Ý Ž h y bj . g x A , p j

js1

F 5 p 5 5h f 5 s h .

Now, n

0 s ² T Ž g . , p: y c² g , p : s ² hg y cg , p : s

¦Ý ;

s

¦

n

;¦

Ý Ž bj y c . g x A , p j

js1

y

n

;

Ž h y c . g xA j , p

js1

Ý Ž h y bj . g x A , p j

js1

G a y b G Ž « y h . ² f , p : y h ) 0, and we have reached a contradiction. Fix b g C and « ) 0, and let g 1 , g 2 g E be such that supp g 1 ; AŽ b, « . and supp g 2 ; V _ AŽ b, « .. We are going to show that 5 g 1 q g 2 5 s max5 g 1 5, 5 g 2 54 . Clearly, we may assume that both AŽ b, « . and V _ AŽ b, « . are of strictly positive m measure. Since E has order semicontinuous norm, it is easy to see that we may also assume that there is d ) 0 such that supp g 2 ;  t: < hŽ t . y b < ) « q d 4 . Denote B1 s  p g Ext E*: aT Ž p . y b F « 4

and

B2 s Ext E* _ B1 .

Now, by Ž)., we have the following. If p g B1 , then < hŽ t . y aT Ž p .< ) d for all t g supp g 2 ; hence ² g 2 , p : s 0. If p g B2 , then < hŽ t . y aT Ž p .< ) < aT Ž p . y b < y « ) 0 for all t g supp g 1; hence ² g 1 , p : s 0. Therefore, max  5 g 1 5 , 5 g 2 5 4 F 5 g 1 q g 2 5 s sup  ² g 1 q g 2 , p : : p g Ext E* 4 s sup  ² g 1 q g 2 , p : : p g B1 4 k sup  ² g 1 q g 2 , p : : p g B2 4 F max  5 g 1 5 , 5 g 2 5 4 . The other assertion follows easily from Proposition 2.1Žb. and the obvious fact that Mh g MultŽ E . for every S M Ž E .-simple function h. COROLLARY 2.5. Let E be a Kothe ¨ function space with order semicontinuous norm. If Mh g MultŽ E . for some nonconstant h g L`Ž m ., then E admits a nontri¨ ial M-projection. The following result furnishes positive answers to Questions 1 and 2 for complex Kothe ¨ spaces.

MULTIPLIERS AND PROJECTIONS

THEOREM 2.6.

93

Let E be a complex Kothe ¨ function space. Then:

Ža. E¨ ery M- or L-projection on E is a characteristic projection. Žb. T g L Ž E . is a multiplier if and only if T s Mh for some h g L`Ž V, S M Ž E ., m .. Proof. Ža. Apply Theorem 1.5 and the description of band projections in Kothe ¨ spaces. Žb. Let T g MultŽ E .. By Corollary 1.4, supp T Ž f . ; supp f for all f g E. Hence, by Proposition 2.2, T s Mh for some h g L`Ž m .. ŽNote that, in particular, every M-projection on E is of the form PA for some A g S.. Thus MultŽ E . ;  Mh : h g L`Ž m .4 . To finish, apply Theorem 2.4. COROLLARY 2.7. A complex Kothe ¨ function space admits a nontri¨ ial M-projection if and only if it admits a nontri¨ ial multiplier. We say that a Kothe space E is semi-rearrangement in¨ ariant Žs.r.i.. if ¨ whenever f g E and g g L0 Ž m . are simple functions, m Žsupp f . - `, and f and g are equimeasurable, then g g E and 5 f 5 s 5 g 5. It is clear that any r.i. space is an s.r.i. space. If E and F are Banach spaces, then we write E ' F when E s F as vector spaces and there is a constant c such that 5 x 5 E s c 5 x 5 F for every x g E. Before starting our next result, let us introduce some additional notation. Given an ideal A in S containing all m-null sets, we shall denote by L`Ž V, A, m . the closure in L`Ž m . of its subspace lin x A : A g A 4 consisting of all A-simple functions. As easily seen, if f g L`Ž m ., then f g L`Ž V, A, m . if and only if for every « ) 0 there exists A g A such that 5 f x A c 5 ` - « , and if and only if  t g V: < f Ž t .< ) l4 g A for every l ) 0. For the particular case of the ideal Sf [  A g S: m Ž A. - `4 , we shall use ˜`Ž m . for L`Ž V, Sf , m ., and let S0 Ž m . stand for the the shorter notation L space of all Sf-simple functions. Also, let SŽ E . [  A g S: x A g E4 . The following lemma will be convenient in the proof of the next theorem. LEMMA 2.8. Assume that the measure space Ž V, S, m . is s-finite and atomless, and let B be a sub-s-algebra of S satisfying the following condition: Ž†. There exist extended real numbers a ) 0 and b ) 0 with a q b s m Ž V . such that B contains e¨ ery set B g S for which m Ž B . s a and mŽ B c . s b . Then B s S.

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Proof. We may assume that a F b . If A g S and m Ž A. - a , then one can easily construct B1 , B2 g S satisfying Ž†. and such that A s B1 l B2 ; hence A g B. Since every set in S can be written as a finite or countable union of such sets A, the assertion follows. THEOREM 2.9. Let E be a s.r.i Kothe ¨ function space o¨ er an atomless measure space Ž V, S, m .. Ža. Žb.

If E admits a nontri¨ ial characteristic L-projection, then E ' L1Ž m .. If E admits a nontri¨ ial characteristic M-projection, then

˜`Ž m . ; E ; L`Ž V , S Ž E . , m . L and there is a constant c ) 0 such that c 5 f 5 ` F 5 f 5 for all f g E

and

˜`Ž m . . 5 f 5 s c 5 f 5 ` for all f g L

In consequence, if E has order semicontinuous norm, then E ' L`Ž V, SŽ E ., m ., and if m Ž V . - ` or E has the Fatou property, then E ' L`Ž m .. Proof. Since supp E s V and m is s-finite, x A g E for every A g S f . It implies that S0 Ž m . ; E. Let E0 denote S0 Ž m . considered as a subspace of E Ži.e., equipped with the induced norm.. Ža. Let PA for some A g S be a nontrivial L-projection on E. Denote by B the subalgebra of S consisting of all B g S for which PB is an L-projection on E0 . We are going to show that B s S. By Lemma 2.8, it is enough to verify that B satisfies condition Ž†. with a [ m Ž A. and b [ m Ž Ac .. Fix any B g S with m Ž B . s a and m Ž B c . s b , and let f g E0 . Thus f s Ý nis1 ri x A i , where ri are scalars and A i g Sf are pairwise disjoint. By the nonatomicity of m , we find B1 , . . . , Bn in S such that m Bi s m A i , m Ž Bi l A. s m Ž A i l B ., and m Ž Bi l Ac . s m Ž A i l B c . for i s 1, 2, . . . , n. Define g s Ý nis1 ri x B i . Then f and g, f x A and g x B , and f x A c and g x B c are equimeasurable. Hence g g E0 and 5 f 5 s 5 g 5 s 5 g x A 5 q 5 g x A c 5 s 5 f x B 5 q 5 f x B c 5. It follows that B g B. Thus for every f g E0 the set function C ª 5 f xC 5 on S is finitely additive. Hence n : Sf ª Rq defined by n Ž C . s 5 xC 5 is a finitely additive measure vanishing on all m-null subsets of V. Moreover, by the nonatomicity of m and the s.r.i. of E, n Ž B . F n Ž C . whenever B, C g Sf and m Ž B . F m Ž C .. In particular, n Ž B . s n Ž C . if m Ž B . s m Ž C .. From this it follows easily that there is a constant c such that n Ž A. s c m Ž A. for all A g Sf .

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Clearly, 5 f 5 s cHV < f < d m s c 5 f 5 1 for every f g E0 . Now observe that, for every V 0 g Sf , the norm of E and the norm of L1Ž m . are both continuous on L`Ž V 0 .. Since the simple functions are dense in L`Ž V 0 ., it follows that 5 f 5 s c 5 f 5 1 for all bounded functions f g E with supp f g Sf . Now, let f g E be arbitrary and choose an increasing sequence Ž V n . in Sf with union V and such that f is bounded on each V n . Then c 5 f x V n 5 1 s 5 f x V 5 F 5 f 5 for every n, hence c 5 f 5 1 F 5 f 5. Thus E ; L1Ž m . and the n norm 5 ? 5 1 is continuous on E. Therefore, 5 ? 5 s c 5 ? 5 1 on the closure E1 of E0 in E. Finally, since S0 Ž m . is dense in L1Ž m ., we must have E1 s L1Ž m .. In conclusion, E s L1Ž m . and 5 ? 5 s c 5 ? 5 1. Žb. Similarly as in Ža. we show that PB is an M-projection on E0 for every B g S. In consequence, 5 f x B j C 5 s max5 f x B 5, 5 f xC 54 for all f g E0 and disjoint B, C g S. It follows that the set function h : Sf ª Rq defined by h Ž C . s 5 xC 5 satisfies h Ž B j C . s maxh Ž B ., h Ž C .4 , whenever B, C g Sf and B l C s B. Moreover, h is related to m in the same manner as n to m in the proof of Ža. above. We now wish to show that there is a constant c such that h Ž C . s c for all C g S f with m Ž C . ) 0. To this end, let B, C g Sf and assume that 0 - m Ž B . F m Ž C .. By nonatomicity, there exist disjoint sets C1 , . . . , Cn g Sf with m Ž Ck . s m Ž B . for k s 1, . . . , n y 1, and m Ž Cn . - m Ž B ., such that CsC1 j ??? j Cn . Then h Ž C . smax 1 F k F n h Ž Ck . smaxh Ž B ., h Ž Cn .4 s h Ž B ., from which the desired property of h is immediate. From what we have just shown it follows that 5 f 5 s c 5 f 5 ` for every f g E0 . By using a similar argument as in Ža., we extend this equality to all bounded functions in E with support in S f , and next obtain c 5 f 5 ` F 5 f 5 for all f g E. Having achieved this, it is easy to conclude the proof. Remarks 2.10. Ž1. The assumption of order semicontinuity in the second part of Theorem 2.10Žb. is essential in case m Ž V . s `. In order to see this, assume m Ž V . s ` and define a lattice seminorm p on L`Ž m . by pŽ f . s inf5 f x A c 5 ` : m Ž A. - `4 . It is easy to verify that pŽ f q g . s pŽ f . k pŽ g . whenever < f < n < g < s 0. Now, declare E to be the space L`Ž m . equipped with the norm given by the equality 5 f 5 s 5 f 5 ` k 2 pŽ f .. Then E is a r.i. Kothe function space with S M Ž E . s SŽ E . s S. Furthermore, ¨ ˜`Ž m . Žs ker p ., but 5 x V 5 s 2 5 x V 5 ` so that 5 f 5 s 5 f 5 ` for all f g L Ž Ž . . E k L` V, S E , m s L`Ž m .. It is clear that the norm of E is not order semicontinuous. Ž2. In Theorem 2.9Žb., if x V g E, then E s L`Ž m . and the norms of these spaces are equivalent. In particular, it is so when E is r.i. and ˜`Ž m .. In general, however, for a s.r.i. space E one may have E/L ˜`Ž m . n E n L`Ž V, SŽ E ., m ., with the norm of E being strictly stronger L than 5 ? 5 ` . All of this, and S M Ž E . s S, occurs in the following example: Assume m Ž V . s ` and fix a disjoint sequence Ž V n . in S with each

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m Ž V n . s `. Let p be the seminorm defined in Ž2. above. For every f g L`Ž m . and n g N set qnŽ f . s npŽ f x V n . ; furthermore, let q Ž f . s sup n qnŽ f .. Finally, define E s  f g L`Ž m . : q Ž f . - `4 , and endow E with the norm 5 f 5 s 5 f 5 ` k q Ž f .. As a consequence of Theorems 1.6 and 2.9Ža., we have the following COROLLARY 2.11. Let E be a s.r.i. Kothe ¨ function space with strictly monotone norm o¨ er an atomless measure space Ž V, S, m .. If there exists a nontri¨ ial L-projection on E, then E ' L1Ž V, S, m .. The next result is a positive answer to Question 3 in the case of complex spaces and is a corollary of Theorems 2.6 and 2.9. THEOREM 2.12. Let E be a complex s.r.i. Kothe ¨ function space o¨ er an atomless measure space Ž V, S, m .. Ža. If there exists a nontri¨ ial L-projection on E, then E ' L1Ž m .. Žb. If there exists a nontri¨ ial multiplier on E and either m Ž V . - ` or E has the Fatou property, then E ' L`Ž m .. Remark 2.13. If m is the counting measure on the space ŽN, 2 N ., then analogues of Theorem 2.9 and Corollary 2.11, concerning symmetric Kothe ¨ sequence spaces, can be obtained by using only elementary means. Consequently, Theorem 2.12 is valid for such sequence spaces as well. The two examples below show that, as far as multipliers and M- or L-projections are concerned, there is a sharp distinction between complex and real Banach lattices or Kothe ¨ spaces. In particular, for real spaces the answers to Questions 1 and 2 are negative. EXAMPLES. Ž1. Consider the real two-dimensional Banach lattices E s l`2 and F s l 12 . Then the operator P defined by P Ž j 1 , j 2 . s 12 Ž j 1 q j 2 , j 1 q j 2 . is an L-projection on E, and an M-projection on F, but in neither case is a band projection. The Kothe ¨ spaces E and F in the next example have similar properties, but are defined over an atomless measure space. Ž2. For every ŽLebesgue. measurable function f on the interval J s wy1, 1x, let the function Rf on J be defined by Ž Rf .Ž t . s f Žyt .. Consider the following real Kothe ¨ function spaces over J, E s  f g L0 Ž J . : 5 f 5 [ 5 < f < q < Rf < 5 ` - ` 4 , F s  f g L0 Ž J . : 5 f 5 [ 5 < f < k < Rf < 5 1 - ` 4 , equipped with the norms indicated in these definitions. Clearly, E and F are just L`Ž J . and L1Ž J . with equivalent norms. In each of these spaces, the formula Pf s 12 Ž f q Rf . defines a bounded linear projection P onto

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the subspace consisting of even functions. Trivially, in neither case is P a band projection. It is easily verified that P is an M-projection on E, and an L-projection on F. Also note that although P is a multiplier on E, it is not a multiplication operator Mh for any h g L`Ž J .. Before we state our next results, recall some auxiliary facts on duality in Kothe function spaces. Given a Kothe function space E, we have E* s ¨ ¨ E* r [ E* s, where E* r and E* s are the closed subspaces Žin fact, bands. consisting of all order continuous Žor integral. and all singular functionals on E, respectively Žsee wK-A, L-Zx.. The operator which assigns to every g g E9 the functional f ª HV gf d m on E is a linear isometry between E9 and E* r. In this way E* r is identified with E9. Note that E* r s E9 is always total on E wK-A, Chap. VI, Sect. 1, Theorem 5x. Below, we shall use the notation Ext* E9 [ E9 l Ext E*. Let us observe that if « g L`Ž m . and < « < s 1, then the multiplication operator M« is an autoisometry of E. It follows that its adjoint M«U maps Ext E* onto itself, and E9 onto E9. In particular, if g g Ext* E9, then also « g g Ext* E9. We shall consider the Žreal or complex. Kothe ¨ function spaces E which satisfy the condition Ext* E9 is total Ž on E . , i.e., separates the points in E.

Ž q.

Under condition Žq., some results for complex spaces remain also valid for real spaces. PROPOSITION 2.14. Let E be a Kothe ¨ function space o¨ er an atomless measure space for which Ext* E9 is total, and let T be a multiplier on E. Then: Ža. supp T Ž f . ; supp f for every f g E. Žb. T Ž f . s aT Ž g . f on supp g for all f g E and g g Ext* E9. In consequence, if g, h g Ext* E9 and < g < n < h < ) 0, then aT Ž g . s aT Ž h.. Proof. Ža. Let f g E and denote B s supp f. It is enough to show that whenever g g Ext* E9 and A s supp g, then T Ž f . s 0 on A l B c. By nonatomicity of the measure, there exists « 1 g L`Ž m . such that < « 1 < s x A l B and ² f, « 1 g : s HV « 1 gf d m s 0. Now, take any « 2 g L`Ž m . with < « 2 < s x A l B c and set « s « 1 q « 2 . Then « g g Ext* E9, ² f, « g : s ² f, « 1 g : s 0, and HA « gT Ž f . d m s ²T Ž f ., « g : s aT Ž « g .² f, « g : s 0. Therefore,

HAlB « c

2

gT Ž f . d m s y

HAlB«

1 gT

Ž f . dm .

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By choosing « 2 appropriately under fixed function « 1 , one can make the integral on the left to be 0 or equal to HA l B c < g < < T Ž f .< d m. But the integral on the right would remain unchanged, hence T Ž f . s 0 on A l B c. Žb. Let f g E and g g Ext* E9. By Ža., for every A g S we have Ž T f x A . s ŽTf . x A and

HAgT Ž f . d m s² x

AT

Ž f . , g: s² T Ž x A f . , g:

s aT Ž g . ² f x A , g : s aT Ž g .

HA gf d m .

It follows that gT Ž f . s aT Ž g . gf, hence T Ž f . s aT Ž g . f on supp g. Let E be a Kothe ¨ function space for which Ext* E9 is total. Define an equivalence relation ; in Ext* E9 as follows: g ; h if and only if there exists a finite sequence Ža chain. g 1 , . . . , g k in Ext* E9 with g 1 s g and g k s h such that < g i < n < g iq1 < ) 0 for i s 1, . . . , k y 1. The resulting partition of Ext* E9 into the equivalence classes modulo ; is countable Žfinite or infinite . and consists of the sets, say, G1 , G 2 , . . . . For every n, let A n [ supp Gn ; that is, A n is a m-minimal set in S containing the support of every g g Gn . The sets A n form a S-partition of V; we shall refer to it as the partition of V determined by Ext* E9. Let us say that Ext* E9 is chainable if g ; h for all g, h g Ext* E9; that is, if the partition determined by Ext* E9 consists only of the set V. It is easy to see that this can be equivalently expressed as follows: There exists a sequence Ž g n . in Ext* E9 such that V s D n supp g n and < g n < n < g nq1 < ) 0 for every n. THEOREM 2.15. Let E be a Kothe function space o¨ er an atomless ¨ measure space for which Ext* E9 is total, and let Ž A n . be the partition of V determined by Ext* E9. Ža. If T g MultŽ E ., then T s Mh for some h g L`Ž m . which is constant on each A n . In particular, e¨ ery M-projection on E is a characteristic projection. Žb. If E has order semicontinuous norm, then there exists a Ž finite or infinite. S-partition Ž Bk . of V, coarser than Ž A n ., with the following property: An operator T g L Ž E . is a multiplier on E if and only if T s Mh , where h is a bounded function on V which is constant on each of the sets Bk . In particular, an operator P g L Ž E . is an M-projection if and only if P s PB , where B is the union of some of the sets Bk . Proof. Ža. Let T g MultŽ E .. By Propositions 2.14Ža. and 2.2, there exists h g L`Ž m . such that T s Mh . By Proposition 2.14Žb., h is constant on each of the sets supp g Ž g g Ext* E9., hence also constant on each of the sets A n .

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Žb. By Ža., every multiplier on E is a multiplication operator. Hence, by Theorem 2.4, MultŽ E . s  Mh : h g L`Ž V, S M Ž E ., m .4 . Moreover, by Ža. and the uniqueness of h in the representations T s Mh , every h g L`Ž V, S M Ž E ., m . is constant on each of the sets A n . It follows that there is a S-partition Ž Bk . of V, coarser than the partition Ž A n ., such that L`Ž V, S M Ž E ., m . consists precisely of those bounded functions on V which are constant on each of the sets Bk . Clearly, Ž Bk . is as required. wThe partition Ž Bk . is formed by the m-maximal sets on which every h g L`Ž m . representing a T g MultŽ E . is constant. Moreover, S M Ž E . is the sub-s-algebra generated by the Bk ’s and the m-null sets.x One of the consequences of Theorem 2.15Ža., is the following. COROLLARY 2.16. Let E be a Kothe ¨ function space o¨ er an atomless measure space for which Ext* E9 is total. Then e¨ ery multiplier on E is tri¨ ial if and only if Ext* E9 is chainable. Two other consequences of Theorem 2.15 are not so straightforward. COROLLARY 2.17. Let the measure space Ž V, S, m . be atomless and separable. If E is an r.i. Kothe ¨ function space with Ext* E9 / B, then e¨ ery multiplier on E is tri¨ ial. Proof. We sketch the proof for the case m Ž V . - `. By a well known wLa, Sect. 14, Theorem 5x, we may assume theorem of Caratheodory ´ without loss of generality that our measure space is the unit circle T ; C with its Haar measure. Let g 1 g Ext* E9. For every s g T, the operator R s defined by Ž R s f .Ž t . s f Ž st . is an autoisometry of E; in consequence, the function RUs Ž g 1 . : t ª g 1Ž sy1 t . is in Ext* E9. Therefore, Ext* E9 is total and chainable, and we finish by appealing to the preceding corollary. function space o¨ er an COROLLARY 2.18. Let E be an s.r.i. Kothe ¨ atomless measure space. If Ext* E9 is total and either E has order semicontinuous norm or m Ž V . - `, then e¨ ery multiplier on E is tri¨ ial. In consequence, an s.r.i. Kothe ¨ function space with order continuous norm has only tri¨ ial multipliers. Proof. Suppose E admits a nontrivial multiplier. Then, by Theorem 2.15Ža., it also admits a nontrivial characteristic M-projection. Hence, by Theorem 2.9Žb., E ; L`Ž m . and there is a constant c ) 0 such that 5 f 5 s c 5 f 5 ` for all f g E. Now, since E contains all simple functions with support of finite measure, it is easy to see that E9 ' L1Ž m .. But Ext L1Ž m . is empty, hence so is Ext* E9; a contradiction. The second assertion follows from the first one and the fact that E* s E9 if Žand only if. E has order continuous norm wK-Ax.

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3. ORLICZ SPACES Let w : Rqª Rq be a nonzero convex function such that w Ž0. s 0. For f g L0 Ž m ., let Iw Ž f . s HV w Ž< f <. d m. The Orlicz space associated to w over the measure space Ž V, S, m . is Lw Ž m . [  f g L0 Ž m . : Iw Ž l f . d m - ` for some l ) 04 , and is equipped with the norm 5 f 5 [ inf « ) 0 : Iw Ž fr« . F 14 . The sequence Orlicz space is usually denoted by lw . It is well known that Lw Ž m . is an r.i. Kothe ¨ function space with the Fatou property. Moreover, the closed subspace of Lw Ž m . consisting of all elements with absolutely continuous norm is equal to Ew Ž m . [  f g L0 Ž m . : Iw Ž l f . - ` for all l ) 04 wK-R, Mu, R-Rx. In wAnx, Ando proved that Lw Ž m . * s Ž Lw Ž m . * r [ Lw Ž m . * s . 1

Ž an l1-direct sum . .

It is known that Ew Ž m .* is linearly isometric to Lw Ž m .* r which can be identified with the Kothe dual Lw Ž m .9. As a consequence of the Ando’s ¨ result we have Ext Lw Ž m .9 ; Ext Lw Ž m .* and thus Ext* Lw Ž m .9 s Ext Lw Ž m .9. Therefore, by the Krein]Milman theorem, Ext* Lw Ž m .9 is total. In view of Corollary 2.18, the above observations lead to the following. THEOREM 3.1. If the measure space Ž V, S, m . is atomless, then e¨ ery multiplier Ž in particular, M-projection. on the Orlicz space Lw Ž m . is tri¨ ial. In order to obtain our next Žand last. result, we first prove the following lemma. LEMMA 3.2. Let g and h be two Ž real or complex . measurable functions on V. Then for e¨ ery constant u 0 ) 0 and A g S with m Ž A. ) 0 one can find a non-null measurable set B ; A and nonzero scalars a and b such that < a < q < b < s 1, and < ag q bh < F u 0 on B. Proof. Let r s u 0r3. It is easy to see that there exist scalars a and b , and a set B ; A with m Ž B . ) 0, such that < g Ž t . y a < F r and < hŽ t . y b < F r for all t g B. Next, there exist nonzero scalars a and b such that < a < q < b < s 1 and < a a q bb < F r. Then, for every t g B, we have < ag Ž t . q bhŽ t .< F < a < < g Ž t . y a < q < a a q bb < q < b < < hŽ t . y b < F 3r s u 0 . THEOREM 3.3. If the measure space Ž V, S, m . is atomless Ž resp. purely atomic. and the Orlicz space Lw Ž m . Ž resp. lw . admits a nontri¨ ial L-projection, then Lw Ž m . ' L1Ž m . Ž resp. lw ' l 1 .. Proof. The proof will be done only for atomless measure Žfor sequence space compare Remark 2.13.. Let P be a nontrivial L-projection on Lw Ž m .. Then P* is a nontrivial M-projection on Lw Ž m .*. The latter space has therefore no nontrivial L-projection which, by the result of Ando mentioned above, implies that Lw Ž m .* s Lw Ž m .* r. In consequence, Lw Ž m .

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s Ew Ž m .. Now, if we assume that w Ž u. ) 0 for all u ) 0, then Lw Ž m . is strictly monotone Žcf. wB-H, Kux. and so Lw Ž m . ' L1Ž m . by Corollary 2.11. Thus it remains to verify that w Ž u. must be positive for all u ) 0. For a contrary, suppose there exists u 0 ) 0 such that w Ž u 0 . s 0 and w Ž u. ) 0 for u ) u 0 . Take arbitrary norm one functions g and h in Lw Ž m . such that Pg s g and Ph s 0. Since g, h g Ew and 5g 5 s 5h 5 s 1 we easily ha¨ e that I w Žg. s I w Žh. s 1 and, for all scalars a and b with
REFERENCES wA-Bx

C. D. Aliprantis and O. Burkinshaw, ‘‘Positive Operators,’’ Academic Press, New YorkrLondon, 1985. wAnx T. Ando, Linear functionals on Orlicz spaces, Nieuw. Arch. Wisk. 8 Ž1960., 1]16. wBe1x E. Behrends, ‘‘M-Structure and the Banach-Stone Theorem,’’ Lecture Notes in Math., Vol. 736, Springer-Verlag, BerlinrHeidelbergrNew York, 1979. wBe2x E. Behrends et al., ‘‘L p-Structure in Real Banach Spaces,’’ Lecture Notes in Math., Vol. 613, Springer-Verlag, BerlinrHeidelbergrNew York, 1977. wB-Hx B. Bru and H. Heinich, Monotonies des espaces d’Orlicz, C. R. Acad. Sci. Paris. Ser. ´ I Math. 301 Ž1985., 893]894. wHWWx P. Harmand, D. Werner, and W. Werner, ‘‘M-Ideals in Banach Spaces and Banach Algebras,’’ Lecture Notes in Math., Vol. 1547, Springer-Verlag, New YorkrBerlin, 1993. wK-Ax L. V. Kantorovich and G. P. Akilov, ‘‘Functional Analysis,’’ Pergamon, Elmsford, New York, 1988. wK-Rx M. A. Krasnoselskii and Ya. B. Rutickii, ‘‘Convex Functions and Orlicz Spaces,’’ Groningen, 1961. wKux W. Kurc, Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation, J. Approx. Theory 69 Ž1992., 173]187. wLx P. K. Lin, L-projections on Banach lattices, Proceedings of the Second Conference on Function Spaces, edited by K. Jarosz, Marcel Dekker, Ž1995., 265]279. wLax H. E. Lacey, ‘‘The Isometric Theory of Classical Banach Spaces,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1974.

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J. Lindenstrauss and L. Tzafriri, ‘‘Classical Banach Spaces, II,’’ Springer-Verlag, BerlinrHeidelberg, New York, 1979. Z. Lipecki, On binary-type approximations in vector lattices Arch. Math. 62 Ž1994., 545]553. W. A. J. Luxemburg and A. Zaanen, ‘‘Riesz Spaces, I,’’ North-Holland, AmsterdamrLondon, 1971. P. Meyer-Nieberg, ‘‘Banach Lattices,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1991. J. Musielak, ‘‘Orlicz Spaces and Modular Spaces,’’ Lecture Notes in Math., Vol. 1034, Springer-Verlag, BerlinrHeidelbergrNew York, 1983. M. M. Rao and Z. D. Ren, ‘‘Theory of Orlicz Spaces,’’ Dekker, New YorkrBaselrHong Kong, 1991. H. H. Schaefer, ‘‘Banach Lattices and Positive Operators,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1974. A. C. Zaanen, Examples of orthomorphisms, J. Approx. Theory 13 Ž1975., 192]204.