Vector Lattice Powers: Continuous and Measurable Vector Functions

Of fundamental importance in the study of order properties of homogeneous polynomials in vector lattices are two constructions: the symmetric positive tensor product , coming from Fremlin [1], and the vector lattice power , introduced by Boulabiar and Buskes [2]; see also the survey paper by Bu and Buskes [3]. Both associate a canonical n -homogeneous polynomial with each Archimedean vector lattice, such that any other homogeneous polynomial of an appropriate class defined on the same vector lattice is the composition of the canonical polynomial with a linear operator. Thanks to this “linearization” various tools of the theory of positive linear operators can be used to study homogeneous polynomials. Thus, the problem of describing the Fremlin symmetric tensor products and the vector lattice powers for

• ^# The research was supported by Russian Science Foundation, project № 24-71-10094, https://rscf.ru/en/ project/24-71-10094/.

• 2.    Vector Lattice Powers

special vector lattices arises. The former enables one to study a large class of order bounded homogeneous polynomials, but has a very complicated structure; the latter has a much more transparent structure, but handles a narrower class of orthogonally additive homogeneous polynomials, see [4–6].

The purpose of this note is to describe the power of the vector lattice of continuous or Bochner measurable vector functions with values in a Banach lattice and to apply this result to the representation of homogeneous orthogonally additive polynomials. We use the standard notation and terminology of Aliprantis and Burkinshaw [7] for the Banach lattice theory, Fremlin [8] for measure theory, Pei-Kee Lin [9] for spaces of measurable vector functions, and the survey paper by Bu and Buskes [3] for order properties of homogeneous polynomials. In the sequel all vector spaces are defined over the field of reals and all vector lattices are assumed to be Archimedean.

Let X be a relatively uniformly complete vector lattice. Then for every positive homogeneous continuous function ^ : RN ^ R and any finite sequence x 1 ,...,x n G X the member ^ ( x i ,...,x n ) G X can be correctly defined by using homogeneous functional calculus, see Definition 3.1 and Theorem 3.7 in [10]. In view of this, we can introduce new vector operations in X by putting x ф y = (x p + y p ) ^1/p and A ® x = A ^1/p x with fixed 0 < p G R, where x,y G X and A G R. In more details, x ф y = %(x,y), where x p : R ² ^ R is defined by ^ _p (r,t) = \ u \^1/p sign u and u = | r | ^p sign r + | t | ^p sign t . Endowed with these new operations, the original order and lattice structure, X becomes a new vector lattice ( X, ф , ® , C ) denoted by X (_p) . For a positive integer p = n G N the vector lattice X (_n) is known as an n-power of X [2].

Assume now that (X, || • ||) is a quasi-Banach lattice (see [5, Definition 2.2]) and define the function || • | (_p) : X (_p) ^ R by | i p( x ) | (_p) := ||x|| ^p ( x G X ). If C is the quasi-triangle constant of the quasi-norm || • | , then ll^{x ф} y ^ ( p ) ^{C 2 |i 1/p|} C ^{p (}ll^xll(p) + llyll(p)) ^{and hence} || • | (_p) is also a quasi-norm; moreover (X (_p) , | • | (_p)) is a quasi-Banach lattice. More details on quasi-Banach space see in [11].

Definition 1. The pair (X (_p) , | • ll( p )) is called the p-concavification of X , see [12] and [4]. Define the canonical order isomorphism i p : X ^ X ( p ) as the identity mapping of the ordered set ( X, C ). If p := n G N, then the mapping j _p : X ^ X ( p ) defined as j _n ( x ):= i _n (x + ) + ( — 1) ⁿ i _n ( x ^- ) ( x G X ) is called the canonical polynomial of X .

Lemma 1. (1) The mapping i p is an odd order isomorphism of X onto X (_p) such that for all x,y G X and A G R the relations hold:

i p (( x p + y ^p ) ^p) = i p( x ) ф i p( y ) , i p ( A ^p x ) = A ® i p( x ) .

• (2)    If X is a ( quasi- ) Banach lattice then X (_p) is a quasi-Banach lattice and i p is a homeomorphism from X onto X (_p) .

• (3)    If n G N then the mapping j _n : X ^ X (_n) is an orthogonally additive n -homogeneous polynomial.

• < 1 See [5, Proposition 3.6]. >

Lemma 2. Let X and Y be uniformly complete vector lattices with respective canonical order isomorphism i x : X ^ X (_n) and i y : Y ^ Y (_n) . Given a lattice homomorphism S : X ^ Y , the mapping S:= iy о S о i X ¹ is a lattice homomorphism from X (_n) to Y (_n) .

• <    Functional calculus in uniformly complete vector lattices commutes with any lattice homomorphism, see [5, Propositions 3.4 and 3.8]. >

It may happen that X ( p ) is not a Banach lattice, even if X is. To avoid this pathology, the additional requirement of p -convexity of X is needed.

Definition 2. A quasi-Banach lattice X is said to be p-convex with 0

от if there exists a constant C such that

1 /p

1 /p

/ m \ 1/p mm       \ 1/p

( E |Xk H    « C (kL ^Xk «J •

for every finite collection { x i ,..., x _m } in X .

Lemma 3. Let X be a ( quasi- ) Banach lattice and 0 < p, q <  от . If X is p -convex and p ^ q then X (_q) is a Banach lattice with respect to some equivalent norm.

• < 1 According to [5, Corollary 3.12], X is p -convex if and only if X (_q) is p/q -convex and if а := p/q ^ 1 then a -convexity implies 1-convexity. By [5, Proposition 2.12] there exists a equivalent Banach lattice norm. >

Definition 3. A mapping P : X ^ Y between vector spaces is called a homogeneous polynomial of degree n (or n - homogeneous polynomial ) if there exists a symmetric n -linear operator P : X ⁿ ^ Y such that P(x) = P ( x,... ,x ) for all x G X . (An n -linear operator ^ : X ⁿ ^ Y is symmetric if ^ ( x i ,...,x _n ) = ^ ( x^ i) ,...,x _a(_n) ) for any permutation a of { 1 , . . . , n } .)

Definition 4. Assume that X is a vector lattice. An n -homogeneous polynomial P : X ^ Y is called orthogonally additive whenever P(x + y) = P(x) + P(y) for any pair of disjoint (i. e. | x | Л | y | = 0) members x,y of X .

Definition 5. Recall that a bornology on a set is a cover of the set that is closed under finite unions and taking subsets. Members of a bornology are called bounded sets . A mapping between bornological spaces is labeled as bounded if it sends bounded sets into bounded sets. A bornology on a vector space is convex if it is stable under the formation of convex hulls. A convex bornological vector space is a vector space equipped with a convex bornology such that addition and scalar multiplication are both bounded.

Denote by P0(ⁿX,Y ) the space of bounded n -homogeneous orthogonally additive polynomials from X to Y and put L ^b (X,Y ) = P 0 ( 1 X,Y ). A vector lattice X is always considered with a bornology containing the totality { [ a, b ] : a, b G X } of order bounded sets.

• Theorem 1. Let X be a uniformly complete vector lattice and Y be a separated bornological vector space. Then for any bounded orthogonally additive n -homogeneous polynomial P : E ^ Y there exists a unique bounded linear operator S : X (_n) ^ Y such that P = T о j _n . Moreover, P^b ( n X, Y ) and L^b (X (_n) , Y ) are linearly isomorphic under the mapping T ^ T о j _n .

• 3.    Spaces of Vector Functions

Remark 1. The result was proved in the author’s paper [13, Theorem 7] for the case of convex bornology; Ben Amor [14, Theorem 26] improved this result showing that the convexity assumption may be omitted. Some special cases of this result were obtained in [3, Theorems 4.3 and 5.4], [15, Theorem 2.4], [16, Theorem 3.3].

In this section we present the necessary information about spaces of strongly measurable and continuous vector functions.

In what follows, Q is an extremally disconnected compactum (i. e. compact Hausdorff topological space) and (Q , E , ^ ) is a measure space that is assumed to be complete and strictly localizable [8, Definition 211E], or, what is the same, to have the direct sum property [17, Section 0.6.4].

Denote by C ^ ( Q,X ) the set of equivalence classes of continuous functions u that act from comeager subsets dom( u ) C Q into X . Recall that a subset of Q is called meager if it is a countable union of nowhere dense subsets of Q and comeager if its complement is meager. Two vector-valued functions u and v are equivalent if u ( q ) = v(q) whenever q G dom( u ) A dom( v ). The set C ^ ( Q, X ) is naturally endowed with the structure of a lattice ordered module over the f -algebra C ^ ( Q ). Observe that C ^ (Q,X ) can be represented also as the set of all extended continuous X -valued functions, see [18, 2.5.1].

Lemma 4. Let Q be an extremally disconnected compact space and let X be a Banach lattice. For every finite collection U i ,..., U n G C ^ ( Q, X ) we have

^(^1, . . .,UN) = ^(ui0, . . . ,UN (-))~, i. e. there exists a comeager subset Qo C Q such that Qo C dom(ui) for all i := 1,..., N, and ^(Ui,... ,Un ) is the equivalence class of the continuous function q H ^(ui (q),... ,un (q)) (q G Qo).

⊳ See [19, Proposition 4.3]. ⊲

Let X be a Banach lattice and let L ⁰ (Q ,X ) := L 0 (Q , S , д, X ) stand for the set of all Bochner measurable functions defined almost everywhere on Q with values in X . Denote by L ⁰ (Q ,X ) := L ⁰ (Q ,X ) / ~ the space of all equivalence classes (of almost everywhere equal) functions from L 0 (Q , X ). The equivalence class of a measurable vector function u G L 0 (Q , X ) will be denoted by й . Define in addition, a scalar multiplication and ordering in L ⁰ (Q ,X ) pointwise, thereby making L ⁰ (Q ,X ) into a vector lattice. A cone of positive vector functions L ⁰ (Q ,X ) + consists of u G L ⁰ (Q ,X ) such that u ( w ) G X + for almost all w G Q.

If и G L ⁰ (Q ,X ), then w H ||u ( w ) || ( w G Q) is a scalar measurable function whose equivalence class is denoted by |u| G L ⁰ (Q).

Definition 6. For an order ideal E in C ^ ( Q ) (resp., in L ⁰ (Q)), assign by definition

E ( X ):= { u G C ^ ( Q,X ) (resp., u G L ⁰ (Q ,X )) : |u| G E } .

If, in addition, ( E, || • | e ) is a Banach lattice, then E(X ) is endowed with the mixed norm defined as ||| u ||| := || |u| || e .

If E = L ^p (Q), then we write L ^p (Q ,X ) instead of E(X ).

Lemma 5. The vector lattice of continuous or measurable vector functions E(X ) is relatively uniformly complete. If, in addition, E is a Banach lattice, then E ( X ) is also a Banach lattices.

• <1 Clearly, if u G L ⁰ (Q ,E ) then the modulus | u | of u coincides with the equivalence class of the vector function w H | u ( w ) | ( w G Q). >

A complete measure space (Q , S ,^ ) is strictly localizable if and only if there exists a linear lifting of L “ ( ^ ), see [8, Sections 341K and 363X(e)]. In this event the quotient algebra B (Q):= S /д ^-1 is complete and its Stone space Q (whose clopen algebra Clop( Q ) is isomorphic to B (Q)) is extremally disconnected. For a fixed lifting p of L “ ( ^ ) there exists a Borel measurable mapping т : Q H Q , called the canonical embedding, such that p = т ^-i о a , where a : B (Q) H Clop( Q ) is the Stone representation, see [17, § 0.6.6 and 0.7.5].

The preimage т ^-i ( V ) of any meager set V C Q is measurable and д -negligible. Denote by т * the mapping which sends each function v G C _M ( Q,X ) to the equivalence class of the measurable function v о т . The mapping т * is a linear and order isomorphism of C _M ( Q,X ) onto L ⁰ (Q , S , д, X ).

Lemma 6. There exists a linear and order isomorphism S from L ⁰ (Q ,X ) onto C ^ (Q,X ) , the Stonian transform of L ⁰ (Q ,X ) . Moreover, |u| = (|S( u )| ◦ т ) " for every u G L ⁰ (Q ,X ) .

• <1 Denote by т the mapping which sends each v G C ^ (Q,X ) to the equivalence class ( v о т ) ~ G L ⁰ (Q , X ). Then S = т ^-1 and u = (S( u ) ◦ т ) ~ for all u G L ⁰ (Q , X ). Details can be found in [17, Theorem 4.2.2]. >

• 4.    Main Results

Since both the continuous and measurable versions of the vector lattice E(X ) are uniformly complete, the expression ^ ( U i ,...,Un ) is well defined for any finite sequence U 1 ,... ,Un G E(X ) and every positively homogeneous continuous function ^ : RN H R. If E is a Banach lattice then E(X ) and E (_n) ( X (_n) ) are equipped with respective mixed norms |||J|| and 111• 111 (Definition 6), while E(X ) (_n) is considered with the power norm 111• 111 ( n ) (Definition 1).

Theorem 2. Let E be an order dense ideal of C ^ ( Q ) and let X be a p -convex Banach lattice with p ^ n . Denote by i e , i x , and I n the canonical order isomorphisms of E , X and E(X ) onto their respective n -powers. Then there exists a \-\-peeserving lattice isomorphism к from E(X) (_n) onto E (_n) (X (_n) ) such that к(^:l _n (-U)) = ( i x о u ) ~ for all u G E(X ) . If, in addition, E is a q -convex Banach lattice with q ^ n , then к is an isometric lattice isomorphism of Banach lattices E(X ) (_n) onto E (_n) (X (_n) ) .

• <    Assume first that E is an order dense ideal of C ^ ( Q ). Define a mapping к from E(X) (_n) onto E (_n) (X (_n) ) by letting к ( v ) be the equivalence class of the vector-function i x о u whenever v = i n ( u ) and u G E(X ). Observe that the vector-function u : dom( u ) ^ X is continuous if and only if so is the vector-function i x о u : dom( u ) ^ X (_n) as i x is a homeomorphism from X onto X (_n) , see Lemma 1. For the same reason, u i ( q ) C u 2 ( q ) for all q G Q o := dom( u _i ) П dom( u ₂ ) if and only if i x о u _i ( q ) C i x ◦ u ₂ ( q ) for all q G Q o and hence u _i and u 2 are equivalent if and only if so are i x о u i and i x о u 2 . It follows that к is a correctly defined order preserving bijection from C ^ ( Q, X ) (_n) into C ^ ( Q, X (_n) ). In fact, к is surjective, since if v G C ^ ( Q,X (_n) ) for some continuous v : dom( v ) ^ X (_n) and u := i x ¹ о v , then by definition u G C_m ( Q, X ) coincides with the equivalence class of i x о u = v , i. e. к( ^ _n U ) = v .

Prove that к is linear. Let vi = in(ui) and v2 = in(u2) for some ui,ui G C^(Q,X). By definition, к(v1 фv2) is the equivalence class of the vector-function ixоu with u = ini(viфv2), so that in(u) = vi ф v2 = in(ui) ф in(u2) = ?n((un + un)i/p).

The last equality is ensured by Lemma 1. It follows that u = ( u n + u n) ^i/n and hence, by Lemma 4, u is the equivalence class of the vector-function q H ( i x о u i )( q ) ф ( i x о u 2 )( q ). The latter coincides with к( v i ) ф к( v 2 ), since the sum in C ^ ( Q,X (_n) ) is defined pointwise, i. e. к( v i ) ф к( v 2 ) is the equivalence class of the vector-function q H i x ( u i ( q )) ф i x ( v i ( q )) ( q G Q ); so к( v i ф v 2 ) = к( v i ) ф к( v 2 ).

Now, if A G R and v = i _n (u) for some u G C ^ (Q,X ) and denote by ® the scalar multiplication in X (_n) , C ^ ( Q,X ) (_n) , and C _M ( Q,X (_nj ). Using again Lemma 1, we see that A ® к( v ) is the equivalence class of the vector function q H A ® i x (u(q)) = i x ( A ^i/n u ( q )). At the same time, A ® v = A ® i _n ( u ) = i _n ( A ^i/n u ) and к( А ® v ) is the equivalence class of the same vector-function by definition, so that к( A ® v ) = A ® к( v ).

It remains to verify that к sends E(X)(n) onto E(n)(X(n)). Taking the definition of the E-valued norm into account, we see that for u G E(X) the following relations hold for all q in some comeager subset of Q:

I к (t n ( u )) |( n ) ( q ) = ^ 1 X u ( q ) \ ( n ) = \\ u(q) \ⁿ = |u| n( q ) -

Hence |u| G E if and only if |u| ⁿ G E (_n) which amounts to saying that u G E(X ) if and only if u ⁿ G E (_n) (X (_n) ). Therefore, к is a norm preserving lattice isomorphism from E(X ) (_n) onto E (_n) (X (_n) ) as claimed. If E is a q -convex Banach lattice and q ^ n, then for u G E(X ) and v = I n ( u ) we have

III к ( v ) III '( n ) = II I* ( v ) 1|1 е _(п) = hl u| " 1Ц. , = ^ I E (I «DH e = 11 u 11 = 11 i nf'S) II = 11 v II W

Thus, we have proved that * is an isometric lattice isomorphism of Banach lattices E(X) (_n) ^{and E} ( n ) ^{( X} ( n ) ). >

Theorem 3. Let EL be an order dense ideal of L ⁰ (Q) and t _n the canonical order isomorphisms of E(X ) onto E(X ) (_n) , whilst X and i x be the same as in Theorem 2. Then there exists a |- (-preserving lattice isomorphism * from E(X) (_n) onto E (_n) (X (_n) ) such that x( i _n ( u )) = ( i x ◦ u ) ~ for all u G E(X) . If, in addition, E is a q -convex Banach lattice with q ^ n , then x is an isometric lattice isomorphism of E(X ) (_n) onto E (_n) (X (_n) ) .

• <    Suppose that IL is an order dense ideal of the vector lattice of measurable functions L ⁰ (Q). Observe first that the vector-function u : Q ^ X is measurable if and only if so is the vector-function i x о u : Q ^ X (_n) and two measurable functions u i ,' 2 : Q ^ R are equivalent if and only if so are i x о u 1 and i x о ' 2 , as i x is a homeomorphism from X onto X (_n) , see Lemma 1. It follows that we can define correctly an order preserving bijection x from L ⁰ (Q , X ) (_n) onto L ⁰ (Q , X (_n) ) by setting x( v ) equal to the equivalence class of the vectorfunction i x о u whenever v = i _n (u) with u G L ⁰ (Q , X ).

To verify that x has the required properties, we use Lemma 6, according to which there are lattice isomorphisms S : L ⁰ (Q) ^ C » ( Q ), S : L ⁰ (Q ,X ) ^ C ^ (Q,X ) and S _n : L°(Q,X (_n) ) ^ C ^ ( Q,X (_n) ) such that |S u| = S (|u|) and |S _n v| = S (|v|) for all u G L ⁰ (Q ,X ) and v G L ⁰ (Q ,X (_n) ). Clearly, E := S(E ) is an order dense ideal in C _M ( Q ) and S ( E (_n) ) = E (_n) . Moreover, |u| G E if and only if |S( u)| G E and |u| G E '_n) if and only if |S( u)| G E (_n) . It follows that S is a lattice isomorphism from E ( X ) onto E(X ) and S _n is a lattice isomorphism from E ' n ) (X (_n) ) onto E (_n) (X (_n) ). By Lemma 2, S is a lattice homomorphism from E ( X ) (_n) onto E(X ) (_n) .

Let к be a lattice isomorphism from E(X ) (_n) onto E (_n) (X (_n) ), whose existence is guaranteed by Theorem 2. To complete the proof, it suffices to show that the following diagram is commutative:

E ( X )---- - E ( X ) ( n )----- E ( n ) ( X ( n ) )

S                  S                    Sn

E(X )--- — E(X ) ( n )---- — E ( n ) ( X ( n ) )

ι n κ

The left square is commutative due to Lemma 2. To prove the commutativity of the right square, take u G E(X) and v G E(X) with v = S(u) or, what is the same, u = (v о т)~.

By sequence using of Lemma 2, Definition of S, к, Sn ¹ , and К we deduce

S _n ¹ о к о S ( t _n ( U ) ) = S _n ¹ о к о I n о S( u ) = S _n ¹ о к о j _n ( v ) = S -1 (( ix о vp ) = ( ix о v о т ) ~ = (ix о up = х (l_n(U)).

It follows that х = Sn ¹ о к о S, whence the required properties of х. О

Definition 7. Let X be a Banach lattice with the canonical polynomial j _n : X ^ X (_n) . Given a vector-function U G L ⁰ (Q ,X ) or U G C ^ (Q,X ), define U n G L ⁰ (Q ,X (_n) ) or U n G C ^ ( Q, X (_n) ) as U n := (uⁿ ) ~ with u ⁿ := j n о u : Q ^ X (_n) . The definition is correct, as j n о u is Bochner measurable or continuous whenever so is u , see Lemma 1 and Definition 1.

Theorem 4. Assume that E is an order dense ideal of L° (Q) or C ^ (Q) . Let X be a uniformly complete vector lattice and let Y be a separated bornological space. Then for any bounded orthogonally additive n -homogeneous polynomial P : E(X ) ^ Y there exists a unique bounded linear operator T : E (_n) ( X (_n) ) H Y such that the representation P(U) = T(u n ) for all U G E(X) . Moreover, P^b (ⁿE(X ) , Y ) and L ^b (E _{( n )} (X _{( n )} ), Y ) are linearly isomorphic under the mapping T H T (( - ) ⁿ ) .

• <    We restrict ourselves to the continuous case E C C ^ ( Q ), since the measurable case E C L ⁰ (Q) is considered in quite a similar way. Assume that к and l _n are the same as in Theorem 2. By Theorem 1, there exist a bounded linear operator S : E(X) (_n) H Y such that P ( U ) = S о j _n (u) for all U G E(X) _n , where j n is a canonical polynomial of the vector lattice E(X ), i. e. j _n (f ) := i _n (f + ) + ( — 1) ⁿ l _n ( f ^- ). If j n : X H X (_n) stand for the canonical polynomial of X , then for every U G E(X ) we have

к( j n( U )) = к( l n( U ⁺ )) + ( - 1) ⁿ к( n_n(u- )) = ( l x о u ⁺G )p + ( - 1) ⁿ ( i x о u ^-0p

= ( j n( u ⁺ ( • ))) - + j n((( - 1)n U ^- (PP = ( j n о up = U n .

It remains to put T := S о к ¹ and observe that P ( U ) = S о j _n ( U ) = T о кj _n ( U ) = T(Uⁿ), as required. ⊲