Operators on injective banach lattices

In this paper we consider some properties of bounded linear operators on injective Banach lattices using a Boolean-valued transfer principle from AL -spaces to injective Banach lattices stated in Kusraev [1]. In Section 2 we collect some Boolean valued representation results for Banach lattices and regular operators (Theorems 2.2, 2.4, and 2.5). In Section 3 we present a Daugavet type equation (Theorem 3.5 and Corollary 3.9) and a Daugavet type inequality (Theorem 3.8) for operators on injective Banach lattices. Section 4 deals with the problem when the spaces of regular (Theorem 4.4), cyclically compact (Theorem 4.7), and cone B-summing (Theorem 4.10) operators are injective Banach lattice.

Recall some basic definitions. A real Banach lattice X is said to be injective if, for every Banach lattice Y , every closed vector sublattice Y 0 ⊂ Y , and every positive linear operator T 0 : Y 0 → X there exists a positive linear extension T : Y → X of T 0 with k T 0 k = k T k . A Dedekind complete AM -space with unit (Abramovich [2] and Lotz [3]) as well as an AL -space (Lotz [3]) is an injective Banach lattice, see Meyer–Nieberg [4].

We denote by P( X ) the Boolean algebra of all band pro jections on a vector lattice X . A crucial role in the structure theory of injective Banach lattice plays the concept of M -projection. A band projection π in a Banach lattice X is called an M -projection if k x k = max {k πx k , k π ^⊥ x k} for all x ∈ X , where π ^⊥ := I X - π . The set M( X ) of all M -projections in X forms a Boolean subalgebra of P( X ). Haydon [5] proved that an injective Banach lattice X is an AL -space if and only if M( X ) = { 0 , I X } .

In what follows X and Y denote Banach lattices, while L ( X, Y ) and L ^r( X, Y ) stand respectively for the spaces of bounded and regular operators from X into Y and k T k _r stands for the regular norm of T ∈ L^r( X, Y ), i. e., k T k r := k | T | k . Throughout the sequel B is

a complete Boolean algebra with unit and zero O, while Λ := Λ(B) is a Dedekind complete AM -space with unit such that B ' P(Λ); in this event B and P(Λ) are identified with taken as the unit both in B and Р(Л). A partition of unity in B is a family ( b ^ ) ^_G = C B such that V g_G = b e = 1 and b e A b_n = O whenever £ = n

For the theory of Banach lattices and positive operators we refer to the books Meyer– Nieberg [4] and Aliprantic and Burkinshaw [6]. The needed information on the theory of Boolean-valued models is briefly presented in Kusraev [7, Chapter 9] and Kusraev and Kutateladze [8, Chapter 1]; details may be found in Bell [9], Kusraev and Kutateladze [10], Takeuti and Zaring [11]. We let := denote the assignment by definition, while N, Q, and R symbolize the naturals, the rationals, and the reals.

• 2.    Boolean Valued Representation

In this section we present some Boolean valued representation results needed in the sequel. Assume that X is a Banach lattice and B is a complete subalgebra of a complete Boolean algebra B( X ) consisting of projection bands and denote by B the corresponding Boolean algebra of band pro jections. We will identify P(Λ) and B.

Definition 2.1. If ( b ^ ) g_G = is a partition of unity in B and ( x ^ ) g_G = is a family in X , then there is at most one element x ∈ X with b ξ x ξ = b ξ x for all ξ ∈ Ξ. This element x , if existing, is called the mixing of ( x ^ ) by ( b ^ ). Clearly, x = o-^2 g_G = b § x ^ • A Banach lattice X is said to be B- cyclic or B- complete if the mixing of every family in the unit ball U ( X ) of X by each partition of unity in B (with the same index set) exists in U ( X ).

A Banach lattice ( X, k · k ) is B-cyclic with respect to a complete Boolean algebra B of band projections on X if and only if there exists a Λ(B)-valued norm · on X such that ( X, · ) is a Banach–Kantorovich space, | x | 6 | y | implies x 6 y for all x, y ∈ X , and k x k = k x k _∞ ( x ∈ X ), see Kusraev and Kutateladze [8, Theorems 5.8.11 and 5.9.1].

Theorem 2.2. A restricted descent of a Banach lattice from the model V ^(B) is a B -cyclic Banach lattice. Conversely, if X is a B-cyclic Banach lattice, then in the model V ^(B) there exists up to the isometric isomorphism a unique Banach lattice X whose restricted descent X ⇓ is isometrically B -isomorphic to X . Moreover, B = M( X ) if and only if [[ there is no M -projection in X other than 0 and I _X ]] = .

C See Kusraev and Kutateladze [8, Theorem 5.9.1]. B

Definition 2.3. The elements X E V ^(B) in Theorem 2.2 and T E V ^(B) in Theorem 2.4 below are said to be the Boolean valued representations of X and T , respectively.

Denote by L_B ^r ( X, Y ) the space of all regular B-linear operators from X to Y equipped with the regular norm k T k _r := inf {k S k : S ∈ LB( X, Y ) , ± T 6 S } . Let X and Y be the Boolean valued representations of B-cyclic Banach lattices X and Y , respectively, while L ^r (X , Y ) stands for the space of all regular operators from X to Y with the regular norm within V ^(B) . The following result states that L ^r (X , Y ) is the Boolean valued representation of L_B ^r ( X,Y ).

Theorem 2.4. Assume that X and Y are B -cyclic Banach lattices, while X and Y are their respective Boolean valued representation. The space L_B ^r ( X, Y ) is order B -isometric to the bounded descent L ^r (X , Y ) ⇓ of L ^r (X , Y ) . The isomorphism is set up by assigning to any T E L B ( X, Y ) the element T := T f of V ^(B) is uniquely determined from the formulas [ T : X → Y ] = and [[ T x = Tx ] = ( x ∈ X ) .

C According to Theorem 2.2 we may assume without loss of generality that X and Y are the bounded descents of some Banach lattices X and Y . Moreover, LB( X, Y ) and

L(X, Y)4 are B-isometric by [7, Theorem 8.3.6]. Since T(X+)T = TT(X+T) = T(X+), it follows that T(X+) C Y+ if and only if [T (X+) C Y+ J = 1. This means that the bijection T ↔ T = T↑ preserves positivity and hence is an order B-isomorphism between LBr (X, Y) and Lr(X, Y)4- Since for S € Lr (X, Y) and S := ST the relations ±T 6 S and [±T 6 S]| = 1 are equivalent, we have [kTkr = |T|rJ = 1, where |T|r = inf{|S| : S € LBr(X, Y), ±T 6 S} and |S| := sup{|Sx| : |x| 6 1}. Thus, it remains to prove that kTkr= k T rk∞ (T ∈ LBr(X, Y)).

If ±T 6 S then || |T|k^ 6 k |S| |U = k S k and hence WT kr > k |T|_r ||— . To prove the reverse inequality take an arbitrary 0 <  e € R and choose a partition of unity ( n ^ ) g_e = in B and a family ( S ^ ) g_e = in L®( X, Y ) such that S ^ > ±T and n ^ |S^| 6 (1 + e ) |T|_r for all £ € E. Define an operator S € L ® ( X, Y ) by Sx := mix g_e = n S ^ x ( x € X ), where the mixing exists in Y , since |S ^ x| 6 (1 + e ) |T |_r |x| and hence ( S ^ x ) is norm bounded in Y . Moreover, Sx = ^Пе n ^ S ^ x in the sense of Л-valued norm on Y . Therefore, S > ±T and |S| 6 (1+ ε ) T r ,

Sx = _ξ π ξ S ξ x in the sense of -valued norm on Y . whence k T k r 6 k S k = klSI Ik 6 (1 + e ) k |Tк |U ▻

Theorem 2.5. Let X be a B -cyclic Banach lattice and let X be its Boolean valued representation in V ^(B) . Then the following hold:

• (1)    V ^(B) к “ X is Dedekind complete” if and only if X is Dedekind complete.

• (2)    VW к “ X is injective” if and only if X is injective.

• (3)    V ^(B) к “ X is an AM -space” if and only if X is an AM -space.

• (4)    V ^(B ) к “ X is an AL -space” if and only if X is injective and B ' M( X ) .

• <1    See Kusraev and Kutateladze [8, Theorems 5.9.6 (1) and 5.12.1]. ▻

• 3.    The Daugavet Equation in Injective Banach Latices

Remark 2.6. As was mentioned in the introduction, Boolean valued analysis approach plays a key role in the proofs below. An alternative approach relies upon Gutman’s theory of bundle representation of lattice normed spaces developed in [12, 13].

Definition 3.1. If X is a real Banach space, a bounded linear operator T : X → X is said to satisfy the Daugavet equation if k I X + T k = 1 + k T k .

Theorem 3.2. If T is a bounded operator on an AL -space X then either T or - T satisfies the Daugavet equation.

• <    The proof and the history of this theorem see in Abramovich and Aliprantis [14, Theorem 11.23], see also Abramovich [15] and Schmidt [16]. ▻

Definition 3.3. Fix a complete Boolean algebra B of band projection in X , i.e., B is a complete subalgebra of P( X ). A bounded linear operator T : X → X is said to satisfiy the Daugavet equation B-uniformly if k π + Tπ k = 1 + k Tπ k for all nonzero π ∈ B. Say that ρ ∈ P( X ) is nonzero over B, whenever πρ 6 = 0 for all nonzero π ∈ B.

Lemma 3.4. Let Λ be a normed lattice with the projection property, X be a decomposable lattice normed space over Λ and k x k := k x k _∞ ( x ∈ X ) . Then for 0 < p ∈ R and x,y € X the inequality |x| > (1 + |y|^p) ^p holds if and only if ||nx || >  (1 + ||ny k ^p ) ^p for all 0 6 = π ∈ P(Λ) .

• <    Prove that ( V n € Р(Л)) ||nx k >  (1 + ||ny k ^p ) ^p implies |x| > (1 + |y |^p) ¹ . If the inequality

|x| > (1 + |y|^p) ^p is not true then there exist a nonzero n g € Р(Л) and 0 <  e € R such that

(1 + e)no|x| < ng(1 + |y|p)p. Notice that |J is P(Л)-homogeneous, i.e., n|x| = |nx| and hence n(l + |y|p)p = (nl + |ny|p)p for all x G X and n G Р(Л), see [8, 5.8.3]. It follows that llnox|| < (1 + E)kno x|| = k(1 + E)nolx| 11^ 6 ||(nol + koyIp) 1 ^ = ||(1 + kn0 ykp) p , a contradiction. The converse implication is immediate from the relations

I nx| = n|x| >  n (l + |y|^p) 1 = ( nl + |ny |^p) ^p , ^ (1 + |y Ip) 1 ^ = (1 + |||y Ir 11^ ) 1 . B

Theorem 3.5. Let X be an injective Banach lattice and an operator T G L ( X ) commutes with all M -projections. Then there exist pair-wise disjoint M -projections n o , n 1 , and П 2 in X such that n o + n i + П 2 = Ix and the operators n i о T + n o о T — П 2 о T and п 1 о T — n o о T — п ₂ о T satisfy the Daugavet equation M( X )-uniformly. Moreover, for any nonzero M -projections p k 6 n k ( k = 1 , 2) the operators — p ₂ о T and p 1 о T fail to satisfy the Daugavet equation.

C Let X , T G V ^(B) be the Boolean valued representations of X and T , respectively. By Theorem 3.3 [X is an AL -space and T G L (X)] = 1. Let the formula ^ ( T ) formalize the sentence ‘ T satisfies the Daugavet equation’ and put П 1 = k(T)], n ₂ = k( — T)], n o = n i о п 2 , and n i = n i — n o . Clearly, n o , n i , and П 2 are pair-wise disjoint. Boolean valued transfer principle together with Theorem 3.1 imply that [[either T or — T satisfies the Daugavet equation]] = 1. It follows from the Transfer Principle that П i V П 2 = k(T) V ^ ( — T)]] = 1, whence n i + n o + n ₂ = 1. Denote by S the mixing of (T , T , — T) by ( n i ,n o , n ₂ ), i. e. n o + n _i 6 [ S = T] and n ₂ 6 [ S = — T ]. If S := S 4 then S := n _i о T + n o о T — п ₂ о T . By applying [7, A.5 (6)] we have n _o + n _i 6 [ ^ (T)] Л [S = T] 6 [ ^ (S)] and n ₂ 6 k(—T)] Л [ S = —T] 6 k(S)] which imply [ ^(S)]| = 1. Since [|S| = |S|] = 1, we have 1 1 +S| = 1+1 S| and taking into account Lemma 3.4 and the easy relation 1 1+A | ^ = 1+||A|k with A G Л yields | n + Sn | >  1 + | Sn | and hence the required equality | n + S | = 1 + | Sn | for all nonzero n G B. The operator n _i о T — n o о T — п ₂ о T is handled similarly. B

We now consider Daugavet type inequalities for regular operators.

For 1 6 p G R and arbitrary s,t G R we denote t ^p := sgn( t ) | t | ^p and a _p ( s, t ) := ( s ^i/p + t ^i/p ) ^p , where 1 /p := p ^-i . In a vector lattice X , we introduce new vector operations ф and * , while the original ordering 6 remain unchanged:

x ф y := a_p ( x, y ) := ( x ^1/p + y ^1/p ) ^p , t * x := t ^p x ( x,y G X ; t G R) .

Then X ^(p) := ( X, ф , * , 6 ) is again a vector lattice. Moreover, ( X ^(p) , | • | _p ) with | x | _p := | х | ^1/р is a Banach lattice called the p -convexification of X , see Lindenstrauss and Tzafriri [17, pp. 53, 54]. Observe that P( X ^(p) ) = P( X ) and M( X ^(p) ) = M( X ). Given a Banach lattice X G V ^(B) and 1 6 p G R, we denote by X ^(p) := X ^{(p A )} the p ^A -convexification of X within V ^(B) . Moreover, if |J and (• |_p are the respective descents (see [8, 1.5.6]) of | • | and | • | _p then I x |_p = |x|^1/p for all x G X .

Theorem 3.6. Let X be an AL -space with a weak order unit and T be a regular linear operator on X ^(p) , 1 6 p G R . Then T ± I x ( p ) if and only if | p ± T p | _r >  (1 + | T p | p ) ^p for all nonzero band projections p in X ^(p) .

C This is a reformulation of the main result (Theorem 9) in Schep [18], since in the case of a function space X we have X ^(p) = { f : | f | ^p G X } . B

To perform the Boolean valued interpretation of Theorem 3.6 we need an auxiliary fact.

Lemma 3.7. Let X be a Banach lattice within V ^(B) . Then for each 1 6 p E R we have

(X ( p ) ) 4 = (X 4 ) ( p ) .

C See Kusraev [19, Lemma 4]. B

Theorem 3.8. Let X be an injective Banach lattice with a weak order unit and an operator T E L ^r ( X ^{( p )} ) commutes with M -projections on X ^{( p )} . The following are equivalent:

• (1)    T ± Ix ( p ) .

• (2)    || np ± nTpU r >  (1 + ||nTp|| p ) p with 0 = п E M( X ) and p E P( X ) nonzero over M( X ) .

• (3)    Up ± Tp | r >  (1 + ||Tp|| p ) p for all nonzero p E P( X ) .

• 4. Injective Banach Latices of Operators

C Let again X E V ^(B) stand for the Boolean valued representation of X . Then X is an AL -space with a weak order unit within V ^(B) by Theorem 2.5, while X ^(p) = (X ^(p) ) 4 by Lemma 3.7. According to Theorem 2.4 there exists a regular operator T E L ^r (X ^(p) ) such that T = T 4 . By Boolean valued transfer principle, Schep’s result (Theorem 3.6) is valid within V ^(B) , i.e., [ T ± I x (p)] = 1 if and only if [ | p ± TpU r >  (1 + | T p | p ) ^{1/p A} for all nonzero band projections ρ on X ]] = . Clearly, [ ρ = 0]] = if and only if πρ = 0 for all nonzero п E M(X). Observe also that |T |_r = || T|| and thus UT | _r = Ц |T|_r ||^ , since an injective Banach lattice is order complete, see [8, Corollary 5.10.7]. It follows that T ± I x ( p ) if and only if |p ± Tp |_r >  (1 +1 Tp |p) p for each p E P( X ) nonzero over M( X ). By Lemma 3.4 the last inequality is equivalent to | np ± nTpU r >  (1 + U nTp U p ) ^p for all nonzero п E M( X ) and p E P( X ) nonzero over M( X ). Thus, (1) ^^ (2), while (3) = ^ (2) is trivial. To prove (2) = ^ (3), take arbitrary nonzero p E P( X ) and put n g := sup { n E M( X ) : np = 0 } , p := p + n g , and n := n ^. Then p is nonzero over M( X ) and Пp = p . Now, making use of (2), we deduce U p ± Tp U r = Щ п /5 ± П Tp|| _r >  (1 + Щ пT/ ^ U p ) p = (1 + U Tp U p ) p . B

The following corollary generalizes Theorem 1 from Shvidkoy [21].

Corollary 3.9. Assume that X is an injective Banach lattice with a weak order unit and an operator T E L ^r ( X ) commutes with all M -projections on X . Then T ± Ix if and only if T satisfy the Daugavet equation P( X ) -uniformly.

C This is immediate from Theorem 3.8 and Proposition 2 in Shvidkoy [21]. B

We consider now under which conditions the space of regular operators between Banach lattices is an injective Banach lattice. First, we state results obtained by Wickstead in [22].

• Theorem 4.1.    If X and Y are Banach lattices, neither of which is the zero space, with Y Dedekind complete then L ^r (X , Y ) is an AL -space under the regular norm if and only if X is an AM -space and Y is an AL -space.

C See Wickstead [22, Theorem 2.1]. B

• Theorem 4.2.    If Y is a nonzero Dedekind complete Banach lattices then L ^r (X , Y ) is an AM -space under the regular norm for every AL -space X if and only if Y is an AM -space with a Fatou norm.

C See Wickstead [22, Theorem 2.3]. B

Denote by K ^r (X , Y ) the linear span of positive compact operators from X to Y endowed with the k -norm defined as UT U k := inf {U S U : ± T 6 S E K (X , Y ) } , see [22].

Theorem 4.3. If X and Y are nonzero Banach lattices, then K ^r (X , Y ) is an AL-space under the k-norm if and only if X is an AM -space and Y is an AL-space.

C See Wickstead [22, Theorem 2.5 (i)]. B

By Boolean valued transfer principle the above three theorems are true within each Boolean valued model. The proofs below are carried out by externalization of these internal facts with X , Y and T standing for Boolean valued representations of X , Y and T , respectively.

Theorem 4.4. Let X and Y be B -cyclic Banach lattices with Y Dedekind complete. Then L_B ^r ( X, Y ) is an injective Banach lattice under the regular norm with B ' M(L_B ^r ( X, Y )) if and only if X is an AM -space and Y is an injective Banach lattice with B ' M( Y ) .

C This a Boolean valued interpretation of Theorem 4.1. According to Theorems 2.4 and 2.5 (4) L_B ^r ( X, Y ) is an injective Banach lattice under the regular norm with B(L_B ^r ( X, Y )) isomorphic to B if and only if L r ( X , Y ) is an AL -space under the regular norm within V ^(B) . Theorem 4.1 (applicable by Theorem 2.5 (1)) tells us that the latter is equivalent to saying that X is an AM -space and Y is an AL -space. It remains to refer again to Theorem 2.5 (3, 4). B

Theorem 4.5. Let Y be a nonzero B -cyclic Dedekind complete Banach lattices. Then L _B ^r ( X, Y ) is an AM -space under the regular norm with M( L _B ^r ( X, Y )) ' B for every injective Banach lattice X with B = M( X ) if and only if Y is an AM -space with a Fatou norm.

C The proof is similar to that of Theorem 4.4: Theorem 4.2 is true within V ^(B) and hence L ^r ( X , Y ) is an AM -space under the regular norm for every AL -space X if and only if Y is an AM -space with a Fatou norm. Moreover, Y has the Fatou norm if and only if [[ Y has the Fatou norm ]] = , see [8, Theorem 5.9.6 (2)]. Now, combining Theorems 2.4 and 2.5 completes the proof. B

Definition 4.6. Denote by Prt(B) (respectively, Prt _σ (B)) the set of all partitions (respectively, countable partitions) of unity in B. A set U in X is said to be mix-complete if, for all ( n ^ ) ^_e n G Prt(B) and ( u ^ ) g_e = C U , there is u 6 U such that u = mix ^_e = nu . Suppose that X is a B-cyclic Banach lattice, ( x _n ) _{n ∈} N ⊂ X , and x ∈ X . Say that a sequence ( x _n ) _{n ∈} N B-approximates x if, for each k ∈ N, we have inf { sup _{n > k} k π _n ( x _n - x ) k : ( π _n ) _n > k ∈ Prt _σ (B) } = 0. Call a set K ⊂ X mix-compact if K is mix-complete and for every sequence ( x _n ) _{n ∈} N ⊂ K there is x ∈ K such that ( x _n ) _{n ∈} N B-approximates x . Observe that if k x k = k x k _∞ ( x ∈ X ) with a Λ(B)-valued norm · , then a sequence ( x _n ) _{n ∈} N in X B-approximates x if and only if inf _n > k x _n - x for all k ∈ N. An operator with values in a B-cyclic Banach lattice is called cyclically compact if (or mix-compact) the image of any bounded subset is contained in a cyclically compact set.

It is clear that in case E = R mix-compactness is equivalent to compactness in the norm topology. Note also that the concept of mix-compactness in Gutman and Lisovskaya [20] coincides with that of cyclically compactness introduced by Kusraev [7], see [20, Theorem 3.4] and [8, Proposition 2.12.C.5].

Given B-cyclic Banach lattices X and Y , denote by KBr(X, Y ) the linear span of positive B-linear cyclically compact operators from X to Y , see [7, 8.5.5]. This is a Banach lattice under the k-norm defined as kTkk:=inf{kSk : ±T 6S∈KBr(X,Y)}.

Note that K ^r ( X, Y ) := K _B ^r ( X, Y ), whenever B = { 0 , } , cp. [22].

Theorem 4.7. Let X and Y be B -cyclic Banach lattices. Then K_B ^r ( X, Y ) is an injective Banach lattice under the k -norm with M(LB ( X, Y )) ' B if and only if X is an AM -space and Y is an injective Banach lattice with M( Y ) ' B .

C The proof runs along the lines of the proof of Theorem 4.4. We have only to observe that an operator T E L B ( X, Y ) is mix-compact if and only if [ T = T f is a compact linear operator from X into Y ] = , see [7, Proposition 8.5.5 (1)]. Thus the B-isometry between L B ( X, Y ) and L^r (X , Y ) 4 induces a B-isometry between K B ( X, Y ) and K ^r (X , Y ) 4 - B

Definition 4.8. Let X be a Banach lattice and Y be a B-cyclic Banach space. Denote by P fin ( X ) the collection of all finite subsets of X . For every T E L ( X, Y ) define

n

n

E ixii i=1

a ( T ):= sup      inf sup E l^k TxB : { x i ,...,X n }E P fin ( X ) ,

( n k ) e Prt _CT ( B ) k_e N =

An operator T E L(X, Y) is said to be cone B-summing if a(T) < to. Thus, T is cone B-summing if and only if there exists a positive constant C such that for any finite collection x1,..., xn E X there is a countable partition of unity (nk )k^N in B with n                  n supV hTxB 6 c Eixii ;

• ^keN i =1                i =1

moreover, in this event a ( T ) = inf {C } . Denote by SB( X, Y ) the set of all cone B-summing operators. The class S B ( X, Y ) was introduced in Kusraev [23, Definition 7.1], see also Kus-raev and Kutateladze [8, 5.13.1]. Observe that if B = { 0 , Iy } then S( X, Y ) := S B ( X,Y ) is the space of cone absolutely summing operators, see Schaefer [24, Ch. 4, § 3, Proposition 3.3 (d)] or (which is the same) 1-concave operators, see Diestel, Jarchow, and Tonge [25, p. 330]. Cone absolutely summing operators were introduced by Levin [26] and later independently by Schlotterbeck, see [24, Ch. 4].

Theorem 4.9. Let X and Y be nonzero Banach lattices. The following are equivalent: (1) S ( X , Y ) is an AL -space.

• (2)    X is an AM -space and Y is an AL -space.

C This result was obtained by Schlotterbeck, see Schaefer [24, Ch. 4, Proposition 4.5]. B Theorem 4.10. Let X be a nonzero Banach lattice and Y be a B -cyclic Banach lattice. The following are equivalent:

• (1)    S B ( X, Y ) is an injective Banach lattice with M( S B ( X, Y )) isomorphic to B .

• (2)    X is an AM -space and Y is an injective Banach lattice with M( Y ) isomorphic to B .

C Suppose that X is a Banach lattice, X is the completion of the metric space X ^∧ within V ^(B) , and Y is the Boolean valued representation of a B-cyclic Banach space Y . Then [ X is a Banach lattice]] = 1 and the map x ^ x ^A is a lattice isometry from X to X 4 -Moreover, for every T E S B ( X, Y ) there exists a unique T := T fE V ^(B) determined from the formulas

[ T E S (X , Y )] = 1, [T x ^A = Tx 1 = 1 ( x E X ) .

The map T H- T is an order preserving B-isometry from SB( X, Y ) onto the restricted descent S (X , Y ) 4 , see Kusraev and Kutateladze [7, 8.3.4] and [8, Theorem 5.13.6]. Note also that X is an AM -space if and only if [[ X is an AM -space ] = by Theorem 2.5 (3). Now, the proof can be carried out in similar lines by Boolean valued interpretation of Theorem 4.9. B

The author thanks the referee for useful remarks leading to improvement of the article.