A characterization of order bounded disjointness preserving bilinear operators

It was observed and employed in [1, 2, 3] that a linear operator T from a vector lattice X to a Dedekind complete vector lattice Y is, in a sense, determined up to an orthomorphism from the family of the kernels of the strata πT of T with π ranging over all band pro jections on Y . Similar reasoning was involved in [4] to characterize order bounded disjointness preserving bilinear operators. Unfortunately, Theorem 3.4 in [4] is erroneous and this note aims to give correct statement and proof of this result. Unexplained terms can be found on the theory of vector lattices and order bounded operators, in [5, 6], on Boolean valued analysis machinery, in [7, 8].

In what follows X , Y , and Z are Archimedean vector lattices, Z ^u is a universal completion of Z , and B : X x Y ^ Z is a bilinear operator. We denote the Boolean algebra of band projections in X by P (X). Recall that a linear operator T : X ^ Y is said to be disjointness preserving if x T y implies Tx T Ty for all x, y € X. A bilinear operator B : X x Y ^ Z is called disjointness preserving (a lattice bimorphism ) if the linear operators B (x, • ) : y H- B(x,y) (y € Y ) and B ( • , y) : x H- B(x,y) (x € X) are disjointness preserving for all x € X and y € Y (lattice homomorphisms for all x € X₊ and y € Y + ). Denote X _n := T { ker(nB( ^ , y)) : y € Y } and Y _n := T { ker(nB(x, • )) : x € X } . Clearly, X _n and Y _n are vector subspaces of X and Y , respectively. Now we state the main result of the note.

Theorem. Assume that X , Y , and Z are vector lattices with Z having the projection property. For an order bounded bilinear operator B : X x Y ^ Z the following assertions are equivalent:

• (1)    B is disjointness preserving.

• (2)    There are a band projection % € P (Z) and lattice homomorphisms S : X ^ Z ^u and T : Y ^ Z ^u such that B(x,y) = %S (x)T(y) — % ^ S (x)T(y) for all (x,y) € X x Y.

• (3)    For every n € P (Z) the subspaces X _n and Y _n are order ideals respectively in X and Y, and the kernel of every stratum nB of B with n € P (Z) is representable as

ker(nB) = [{XT x Y _T : ст, т € P (Z); ст V т = n}.

The proof presented below follows along general lines of [1–4]: Using the canonical embedding and ascent to the Boolean valued universe V ^(B) , we reduce the matter to characterizing disjointness preserving bilinear functional on the product of two vector lattices over dense subfield of the reals R . The resulting scalar problem is solved by the following simple fact.

Lemma 1. Let X and Y be vector lattices. For an order bounded bilinear functional в : X x Y ^ R the following assertions are equivalent:

• (i)    β is disjointness preserving.

• (ii)    ker(e) = ( X q x Y ) U (X x Y q ) for some order ideals X q С X and Y q C Y.

• (iii)    There exist lattice homomorphisms g : X ^ R and h : Y ^ R such that either в(x,y) = g(x)h(y) or в (x,y) = - g(x)h(y) for all x E X and y E Y.

C Assume that ker(в) = ( X q x Y ) U (X x Y q ) and take y E Y . If y E Y q then в( • , y) = 0, otherwise ker(в( • , y)) = X q and в( • ,y) is disjointness preserving, since an order bounded linear functional is disjointness preserving if and only if its null-space is an order ideal. Similarly, в(x, • ) is disjointness preserving for all x E X and thus (ii) = ^ (i). The implication (i) = ^ (iii) was established in [9, Theorem 3.2] and (iii) = ^ (i) is trivial with X q = ker(g) and Y q = ker(h). B

Let B be a complete Boolean algebra and V ^(B) the corresponding Boolean valued model with Boolean truth values [ y] for set-theoretic formulas y. There exists an element R E V ^(B) which plays the role of a field of reals within V ^(B) . The descending functor sends every internal algebraic structure A into its descent A^ which is an algebraic structure in conventional sense. Gordon’s theorem (see [5, 8.1.2] and [10, Theoren 2.4.2]) tells us that the algebraic structure R ^ (with the descended operations and order relation) is an universally complete vector lattice. Moreover, there is a Boolean isomorphism x of B onto P ( R ^ ) such that b 6 [ x = y ] if and only if x(b)x = x(b)y. We identify B with P ( R ^ ) and take x to be I b .

Let [X x Y, R ^ ] E V and [ X ^л x Y ^л , R ] E V ^(B) stand for the sets respectively of all maps from X x Y to R ^ and from X ^л x X ^л to R (within V ^(B) ). The correspondences f ^ f t , the modified ascent, is a bijection between [X x Y, R ^ ] and [X ^л x Y ^л , R ]. Given f E [X, R ^ ], the internal map f t E [ X ^л , R ]] is uniquely determined by the relation [ f t (x ^л ) = f(x )] = 1 (x E X). Observe also that n 6 [ f t (x ^л ) = nf(x)] (x E X, n E P ( R ^ )). This fact specifies for bilinear operators as follows.

Lemma 2. Let B : X x Y ^ Y be a bilinear operator and в := B t its modified ascent. Then в : X ^л x Y ^л ^ R is a R ^л -bilinear functional within V ^(B) . Moreover, B is order bounded and disjointness preserving if and only [[ β is order bounded and disjointness preserving ] = .

C The proof goes along similar lines to the proof of Theorem 3.3.3 in [10]. B

Lemma 3. Let B and β be as in Lemma 2 . Then [ ker(B) ^∧ = ker(β) ] = .

C Using the above mentioned determining property of modified ascent and interpreting the formal definition z E ker(в) о ( 3 x E X ^л )( 3 y E Y ^л )(z = (x, y) Л в(x,y) = 0), the proof is reduced to a straightforward calculation:

[ z E ker(в)] =   _    [ z = (x ^л ,y ^л ) Л в(x ^л ,У ^л ) = 0]

x E X,y E Y

= V [z = (x, уГ Л (x, у ) ^л E ker(B) ^л]

( x,y ) e X x Y

6 [ z G ker(B) ^Л] = V    [ z = (x, у ) ^{Л Л} ( x-У ) G ker( ^B)l

( x,y ) E X x Y

= V ^[(z = ⁽х ^{Л ,}У ^{Л ) Л в(}x ^{Л ,}У ^{Л )} = ⁰] x E X,y E Y

6 [ z G ker(e)]• B

Lemma 4. Define X and Y within V ^(B) by X := T { ker(e( • , Y)) : y G Y ^Л } and Y := T { кег(в(x, • )) : x G X ^Л } . Given arbitrary п G P (Z), x G X, and y G Y, the equivalences hold:

n 6 [ x ^л G X] < >  x G X _n , n 6 [ у ^Л G Y ] <  >  y G Y _n .

C For n G P ( Z ) and x G X we need only to calculate Boolean truth values taking into account that [ B(x,y) = в(x ^л ,v ^л)]] = 1 for all x G X and y G Y :

[ x ^л G X ' = [ ( V v G Y ^Л )в(x ^л ,v) = 0] = Д [ в(x ^л ,v ^л ) = 0] = Д [ B(x,v) = 0]. v E Y                    v E Y

It follows that n 6 [ x ^л G X] if and only if п 6 [ B(x,v) = 0] for all v G Y . By Gorgon’s theorem the latter means that nB (x, v) = 0 for all v G Y , that is x G X _n . B

Lemma 5. Let B and в be as in Lemma 2. For arbitrary п G P (Z), x G X , and y G Y, we have п 6 [ (x ^л ,y ^л ) G ( X x Y ) U (X x Y )] if and only if there exist а,т G P (Z) such that а V т = п, x G X _a , and y G Y _T .

C Denote p:= [(xл,yл) G (X x Y) U (X x Y)] and observe that p = [(xл G X) V ул G Y1 = [xл G XJ V [ул G YJ.

Clearly, n 6 p if and only if а V т = п for some а 6 [ x ^л G X] and т 6 [ у ^л G Y ], so that the required property follows from Lemma 4. B

Proof Of the main result. The implication (1) = ^ (2) was proved in [9, Corollary 3.3], while (2) = ^ (3) is straightforward. Indeed, observe first that if (2) is fulfilled then | B(x,y) | = | B | ( | x | , | y | ) = | S | ( | x | ) | T | ( | y | ), so that we can assume S and T to be lattice homomorphisms, as in this event ker(B) = ker( | B | ). Take п G P (Z) and denote а := п — n[Sx] and т := п — n[Ty], where [y] is a band projection onto { y } ^±± . Observe next that nB ( x,y ) = 0 if and only if n[Sx] and n[Ty] are disjoint or, what is the same, if а V т = п. Moreover, the map p _y : x ^ aS (x)T(y) is disjointness preserving for all y G Y and hence X „ = Q_{y E Y} ker(p _y ) is an order ideal in X. Similarly, Y _T is an order ideal in Y . Thus, (x, y) G ker(nB) if and only if x G X ^ and y G Y _T for some а,т G P (Z) with а V т = п.

Prove the remaining implication (3) = ^ (1). Suppose that for every п G P (Y) the representation in (3) holds. Take x,u G X and put п := [ x ^л G X ], p := [ | и | ^Л 6 | x | ^л ]. By Lemma 4 we have x G X _n . Note also that either p = 0 or p = 1 . If p = 1 then | u | 6 | x | and by hypotheses u G X _n . Again by Lemma 4 we get p 6 [ и ^Л G X ]. This estimate is obvious whenever p = 0 , so that [ x ^л G X] Л [ | и | ^Л 6 | x | ^л] ^ [ и ^Л G X] = 1 for all x,u G X. Now, a simple calculation shows that X is an order ideal in X ^Л :

[ ( V x,u G X ^Л )( | и | 6 | x | Л x G X ^ u G X )]

= Д ( [ x G X] Л [ | u | 6 | x |] ^ [ u G X ]) = 1 . u,x E X

Similarly, Y is an order ideal in Y ^∧ .

It follows from the hypothesis (3) and Lemma 5 that (x, y) G ker(nB) if and only if n 6 [(х ^л ,у ^л ) G ( X x Y ) U (X x Y )]]. Taking into account Lemma 2 and the observation made before it we conclude that n 6 [(х ^л ,у ^л ) G ker(e)]) if and only if n 6 [(х ^л ,у ^л ) G ( X x Y ) U (X x Y )] and hence [ ker(e) = ( X x Y ) U (X x Y )] = 1. It remains to apply within V ^(B) the equivalence (i) ^^ (iii) in Lemma 1. It follows that B is disjointness preserving according to Lemma 2. B

Corollary. Assume that Y has the projection property. An order bounded linear operator T : X ^ Y is disjointness preserving if and only if ker(bT) is an order ideal in X for every projection b G P (Y) .

C Apply the above theorem to the bilinear operator B : X x R ^ Y defined as B (x, A) = AT(x) for all x G X and A G R . B