Positivity 11 (2007), 201–230 c 2007 Birkh¨
auser Verlag Basel/Switzerland 1385-1292/020201-30, published online April 6, 2007 DOI 10.1007/s11117-006-2023-0

Positivity

On the Theory of Lp(Lq )-Banach Lattices C. Ward Henson and Yves Raynaud Abstract. Given 1 ≤ p, q < ∞, let BLp Lq be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some Lp (Lq )Banach lattice. We show that the range of a positive contractive projection on any BLp Lq -Banach lattice is itself in BLp Lq . It is a consequence of this theorem and previous results that BLp Lq is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BLp Lq Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BLp Lq . We also consider the class of all sublattices of Lp (Lq )-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of Lp -Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞. Mathematics Subject Classification (2000). Primary: 46B42, 46E30; Secondary: 03C65, 46M07. Keywords. Banach lattices, Lp-spaces.

Introduction An Lp -Banach lattice is one that is isometrically lattice isomorphic to Lp (Ω, Σ, µ) for some measure space (Ω, Σ, µ). A well-known characterization of Lp -Banach lattices was given in the early 1940s by Kakutani (when p = 1) and Bohnenblust (when 1 < p < ∞). A Banach lattice X is called an abstract Lp -space if the pth power of its norm is disjointly additive; i.e., if for every pair of disjoint elements x and y in X we have: x + yp = xp + yp . The Kakutani-Bohnenblust characterization states that the class of Lp -Banach lattices coincides with that of abstract Lp -spaces. (Bohnenblust’s proof in the case p = 1 required some additional hypotheses that were removed by Nakano and Ando; see the historical discussion in [L], §5.15. A short proof can be found in [LT],

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Theorem 1.b.2.) In other words the class of Lp -Banach lattices is axiomatizable in the class of all Banach lattices by a single statement, namely: 
(0.1) ∀x∀y |x| ∧ |y| = 0 =⇒ x + yp = xp + yp . In the present paper we seek to characterize the more complicated class of Lp (Lq )Banach lattices in a similar way. Indeed, we seek axioms in the form of firstorder statements in the formal language appropriate for Banach lattices; these are finite-length, closed formulas involving the vector lattice operations, the norm, and first-order quantifiers (i.e., only those ranging over elements of the space). A precise description of an admissible form of such axioms is given in the monograph [HI]; they are positive bounded sentences of the appropriate first-order language, interpreted using an approximate notion of satisfaction. A remarkable connection between this logic and the theory of Banach space ultraproducts, stated in [HI] as Proposition 13.6, is the following: a class (of Banach lattices, say) is axiomatizable by a set of positive bounded sentences iff it is closed under ultraproducts and the complementary class is closed under ultrapowers (up to isometric lattice isomorphisms). Note that this connection does not provides a comprehensible set of axioms. It does provide an explicit but very large one, namely the set of all sentences which are satisfied by all the members of the class under consideration (known in logic as the theory of the class). A description of ultraproducts of Lp (Lq )-Banach lattices has been known since the 1980s. This class is not closed under ultraproducts, but a somewhat larger class is, namely the class of bands of Lp (Lq )-Banach lattices ([LR1],[HLR]). An “abstract” description of these bands as Banach lattices has been given. By an abstract Lp (Lq )-space (in short, an ALp Lq -space) we mean a Banach lattice X which, for some measure space (Ω, Σ, µ), can be equipped with the structure of an L∞ (Ω, Σ, µ)-module and with a map N : X → Lp (Ω, Σ, µ)+ such that: (i) For every ϕ ∈ L∞ (Ω, Σ, µ) and x ∈ X, if ϕ ≥ 0 and x ≥ 0 then ϕ.x ≥ 0; (ii) N (x + y) ≤ N (x) + N (y) for every x, y ∈ X; (iii) N (ϕ.x) = |ϕ|N (x) for every ϕ ∈ L∞ (Ω, Σ, µ) and x ∈ X; (iv) if 0 ≤ |x| ≤ |y| then N (x) ≤ N (y), for every x, y ∈ X; (v) N (x + y)q = N (x)q + N (y)q for every disjoint pair x, y ∈ X; (vi) xX = N (x)Lp for every x ∈ X. (These spaces were called “random abstract Lq -spaces over Lp ” in the monograph [HLR], (see chap. 8); the more concise terminology used here seems preferable for this paper.) A map N verifying the axioms (i)–(vi) will be called a q-random norm. The class of ALp Lq -spaces coincides with that of bands in Lp (Lq )-Banach lattices equipped with the natural N -map: f (·) → f (·)Lq . (See [LR1] or [HLR], Theorem 8.7.) This provides a characterization of bands in Lp (Lq )-spaces as the Banach lattices arising from ALp Lq -spaces by ignoring the action of L∞ and the random norm N . (As noted above, we will call these BLp Lq -Banach lattices.) This characterization involves properties of mathematical objects which do not belong

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to the lattice nor to the scalar field (the measure space (Ω, Σ, µ) and the map N ); thus it cannot provide a set of axioms in the language of Banach lattices. However, we prove in this paper that the class of BLp Lq -Banach lattices is axiomatizable in the class of Banach lattices. Our first proof is by verifying that the complementary class of Banach lattices is closed under ultrapowers; equivalently, we show that the class itself is closed under ultraroots. (An ultraroot of a Banach lattice X is a Banach lattice Y having an ultrapower YU that is isometrically lattice isomorphic to X). The main ingredient for obtaining this result is an extension of a wellknown result of Douglas and Andˆ o (about contractively complemented subspaces of Lp -spaces) to the setting of Lp (Lq )-Banach lattices: we prove in section 2 that a contractively complemented sublattice of a BLp Lq -Banach lattice is again a BLp Lq -Banach lattice. For obtaining a more intrinsic characterization of BLp Lq -Banach lattices in the language of Banach lattices, we define in section 3 a notion of “scriptLp (Lq )-lattice” (for which we introduce the term: Lp Lq -lattice), which is analogous to the familiar Banach space notion of Lp -space, and we show that the class of (Lp Lq )1 -lattices (the analogues of Lp,1 -spaces) coincides with that of BLp Lq Banach lattices. This permits us to give an explicit set of axioms for the class of BLp Lq -Banach lattices. Finally, in section 4 we give a characterization of sublattices of Lp (Lq )-Banach lattices. Since this class is closed under ultraproducts and sublattices, it is axiomatizable by positive bounded first-order sentences. We give an explicit set of axioms (in the case where p/q is not an integer) inspired by the axioms given in the 1960s by J.-L. Krivine for the class of subspaces of Lp -Banach spaces [K2]. In the case p < q we obtain our characterization using the theory of negative definite functions on abelian semi-groups, as exposed in [BCR]. The characterizing condition states that the pth -power of the norm is negative definite on the positive cone of the Banach lattice, with respect to the semi-group operation (x, y) → (xq + y q )1/q . This is to be compared with the classical characterization of the subspaces of Lp (for p ≤ 2) by the condition that the pth -power of the norm is negative definite on the Banach space, with respect to the underlying additive group structure (see [BDK]). We also give a characterization of sublattices in the case q = ∞. For clarity of expression, we often write explicit, easy to understand axioms in this paper that are not, strictly speaking, positive bounded sentences in the sense of [HI]. For example, the statement (0.1) expressing the pth -power additivity of the norm in a Banach lattice is not a positive bounded sentence. (It’s quantifiers are not bounded, it involves an implication and hence is not positive, and the equation x + yp = xp + yp is not an atomic formula in the language of Banach lattices.) In this paper we are going to omit the arguments needed to translate such axioms into the language of positive bounded sentences, since including them would distract from the main points of the paper without adding much information. A convenient framework in which to formulate such axioms is

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the emerging continuous version of first-order logic [BBHU, BU]; this is equivalent in expressive power to the logic of positive bounded sentences [HI].

1. Preliminaries Maharam and decomposable measure spaces. ([F]) In the following we consider only Maharam measure spaces. A measure space (Ω, Σ, µ) is Maharam iff it is semi-finite (i.e., every set A ∈ Σ with µ(A) > 0 contains a subset B ∈ Σ with 0 < µ(B) < ∞) and the space L∞ (Ω, Σ, µ) is ordercomplete (i.e., every order bounded family of elements has a l.u.b.; equivalently, L0 (Ω, Σ, µ) is order-complete). An equivalent condition for a semi-finite measure space (Ω, Σ, µ) to be a Maharam space is that the dual of L1 (Ω, Σ, µ) is identifiable with L∞ (Ω, Σ, µ) via the pairing given by the integral. Every σ-finite measure space is Maharam. Since we are considering here non σ-finite measure spaces (e.g., measure spaces associated with abstract L1 -spaces such as ultrapowers of L1 -spaces) the following generalization is natural.  A measure space (Ω, Σ, µ) is decomposable if there exists a partition Ω = Ωα α

of Ω into elements Ωα of Σ which have finite nonzero µ-measure, such that (i) a subset A of Ω belongs to Σ if and only  if each Ωα ∩ A belongs to Σ (ii) and in that case one also has µ(A) = µ(A ∩ Ωα ). α

We shall call such a decomposition a basic decomposition. Every Lp -space (1 ≤ p < ∞) can be represented as the Lp -space of a decomposable measure space. Decomposable measure spaces have interesting features (see [F]). In particular, a decomposable measure space is a Maharam space; conversely, for every Maharam measure space (Ω, Σ, µ), the spaces L∞ (Ω, Σ, µ), resp. L0 (Ω, Σ, µ) can be represented as L∞ (Ω , Σ , µ ), resp. L0 (Ω , Σ , µ ), where (Ω , Σ , µ ) is decomposable. If A, B ∈ Σ we say that A is µ-almost included in B (notation A ⊂µ B) if of meaµ(A\B) = 0; equivalently, 1A ≤ 1B in L∞ (Ω, Σ, µ). If (Aα ) is a family  surable sets in a Maharam measure space (Ω, Σ, µ) we denote by Aα a least α  upper bound of the family (Aα ); this is any set A ∈ Σ such that 1A = 1Aα . α

Equivalently, A is a least upper bound iff Aα ⊂µ A for every α, and whenever A is another set in Σ having this property, then A ⊂µ A . (By standard abuse  of language we speak of “the” µ-least upper bound). We define analogously Aα , α

the greatest lower bound of the family (Aα ); we say that two sets A, B ∈ Σ are µ-disjoint if A ∧ B ⊆µ ∅; equivalently there are disjoint sets A , B  in Σ which are µ-equivalent to A, resp. B (µ(AA ) = 0 = µ(B B  )).

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If (Ω1 , Σ1 , µ1 ) and (Ω2 , Σ2 , µ2 ) are σ-finite measure spaces it is usual to denote by Σ1 ⊗ Σ2 the σ-algebra of subsets of Ω1 × Ω2 generated by the set R of rectangles A1 × A2 , where A1 ∈ Σ1 and A2 ∈ Σ2 , and by µ1 ⊗ µ2 the unique σ-additive measure µ on Σ1 ⊗ Σ2 such that µ(A1 × A2 ) = µ1 (A1 )µ2 (A2 ) for every A1 ∈ Σ1 , ¯ 2 defined by A2 ∈ Σ2 . In the general case consider the larger σ-algebra Σ = Σ1 ⊗Σ taking Σ to be the set of all subsets E of Ω1 × Ω2 such that A2 1 ∀A1 ∈ Σ1 , A2 ∈ Σ2 of finite measure, (A1 × A2 ) ∩ E ∈ ΣA 1 ⊗ Σ2 i where ΣA i is the trace of Σi on Ai . If E ∈ Σ, then we set

µ1 ⊗ µ2 (E) = sup{µ1|A1 ⊗ µ2|A2 ((A1 × A2 ) ∩ E) | Ai ∈ Σi , µi (Ai ) < ∞}. ¯ 2 , µ1 ⊗ Then the product of the two measure spaces is defined to be (Ω1 ×Ω2 , Σ1 ⊗Σ µ2 ). A product of decomposable measure spaces is itself decomposable. Indeed, if (Ω1,α ) and (Ω2,β ) are basic decompositions of Ω1 , resp. Ω2 into finite-measure disjoint elements, then (Ω1,α × Ω2,β )α,β is a basic decomposition for the product measure space. (It is not hard to see that the completion of the measure space (Ω1 × Ω2 , Σ, µ1 ⊗ µ2 ) coincides with the “c.l.d. product measure space” of [F], Definition 251F). Complex Banach lattices Complex Banach lattices are usually defined as complexifications of real Banach lattices (e.g., see [MN], §2.2). We shall adopt a slightly more axiomatic point of view. A complex Banach lattice is a complex Banach space equipped with a norm-preserving antilinear involution x → x ¯ (the ‘conjugate’ of x) and a map x → |x| (the ‘modulus’) such that (i) for every x ∈ X, |x| = |x|; | |λx| = |λ||x|; (ii) for every x ∈ X and λ ∈ C, (iii) the restriction of the modulus to Re X := {x ∈ X | x = x ¯} (which is a real linear subspace of X) gives it the structure of a real vector lattice;  ¯)/2; (iv) for every x ∈ X, |x| = (cos θ|Re x| + sin θ|Re ix|), where Re x := (x + x θ

(v) for every x, y ∈ X such that |x| ≤ |y| one has x ≤ y. Note that, in particular, Re X is a real normed Banach lattice, and that in the notation of [LT], axiom (iv) can be written |x| = (|Re x|2 + |Re ix|2 )1/2 . An isomorphism of complex Banach lattices is an isometric linear map preserving the conjugate and modulus functions. K¨ othe function spaces. ([HLR]) If (Ω, Σ, µ) is a (semi-finite) measure space, we denote by L0 (Ω, Σ, µ) the space of measurable (real or complex) scalar-valued functions defined on Ω, modulo equality µ-almost everywhere. This is a topological Riesz space (for the topology of convergence in measure). A K¨ othe function space on (Ω, Σ, µ) is a linear subspace X such that: (i)

X is an order ideal of L0 (Ω, Σ, µ) (i.e., it is a linear subspace and it is solid: if f ∈ X, g ∈ L0 , and |g| ≤ |f |, then g ∈ X);

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X is equipped with a norm for which it is complete and which is order-compatible (i.e., if f, g ∈ X and |g| ≤ |f |, then g ≤ f ); X is order-dense in L0 (Ω, Σ, µ) (i.e., every A ∈ Σ with µ(A) > 0 contains B ∈ Σ, such that µ(B) > 0 and the indicator function 1B belongs to X).

This definition generalizes slightly the one given in [LT], Definition 1.b.17. It is preserved under changes of density and applies to not necessarily σ-finite measure spaces. Recall that a Banach lattice X is order continuous if xα  → 0 for every downward directed net (xα ) of positive elements of X whose g.l.b. is 0. Every order continuous Banach lattice can be represented as a K¨ othe function space on a decomposable measure space. Moreover, if X is an order-continuous K¨ othe function space over a Maharam measure space, its dual is also a K¨ othe function space over the same measure space. Indeed, its dual is the associated function space (or K¨ othe dual) X  = {f  ∈ L0 (Ω, Σ, µ) | f  g ∈ L1 (Ω, Σ, µ) for every g ∈ X}; the pairing for this duality is the usual integral pairing given by:  f  , g = f  g dµ. In an order continuous K¨ othe function space X the support of every element f is σ-finite. (Proof: otherwise one could find an uncountable family of pairwise disjoint nonzero elements of the form fα = 1Aα f . Some infinite subfamily (fα )α∈Γ would be bounded from below: fα  ≥ r for some rational number r > 0. This subfamily would generate a closed sublattice linearly and lattice isomorphic (but not necessarily isometric) to ∞ (Γ). But such a Banach lattice is not order continuous, so this is a contradiction.) Lp (Lq )-spaces If 1 ≤ p < ∞ and Y is a Banach space, then Lp (Ω, Σ, µ; Y ) is the space of (classes of) Bochner-measurable functions f : Ω → Y such that (the class of) the function NY (f ) : Ω → IR+ , ω → f (ω)Y belongs to Lp (Ω, Σ, µ) (recall that Bochner-measurable functions are limits of sequences of measurable simple functions). If Y itself is a space Lq (Ω , Σ , ν  ), 1 ≤ q < ∞, then Lp (Ω, Σ, µ; Y ) can be identified with a K¨ othe function space X on the product measure space ¯  , µ⊗µ ¯  ): an element f ∈ L0 (Ω × Ω , Σ⊗σ ¯  , µ⊗µ ¯  ) belongs to X iff it (Ω × Ω , Σ⊗Σ has a representative f˜ which is countably supported on the basic decomposition (Ω1,α ×Ω2,β )α,β (hence f˜ is measurable for the smaller σ-field Σ1 ⊗Σ2 ), the (classes of the ) partial functions f˜(ω, ·) of which belong to Lq (Ω , Σ , ν  ) for every ω ∈ Ω  and the (class of the) function Nq (f ) : Ω → IR+ , ω → ( |f˜(ω, ω  )|q dµ (ω  ))1/q belongs to Lp (Ω, Σ, µ). Abstract Lp (Lq )-spaces Abstract Lp (Lq )-spaces (where 1 ≤ p, q < ∞) were defined in the Introduction. The ordinary Bochner spaces of (classes of) Lq -valued p-integrable functions are clearly ALp Lq -spaces (the random norm was denoted by Nq at the end of the previous paragraph); in particular, Lp -spaces and Lq -spaces are ALp Lq -spaces. In

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fact, if the Lp -space is 1-dimensional then the definition of ALp Lq -space coincides with that of ALq -space (abstract Lq -space), see [LT] or [L]. More generally, any band B in a Bochner Lp (Ω, Σ, µ; Lq )-space inherits the structure of an ALp Lq space . Conversely it was proved in [LR1] (see also [LR2] and [HLR]) that every ALp Lq -space is isometrically lattice isomorphic to a band of a Bochner Lp (Lq )space. In fact the isomorphism preserves the action of L∞ and the q-random norm. In the separable case, every ALp Lq -space X has a concrete representation as a p-direct sum of Bochner Lp (Lq )-spaces; namely, X is isometrically lattice isomorphic to a space of the form:  



n  n Lp (Ωn ; q ) ⊕ Lp (Ω∞ ; q ) ⊕ Lp (Ωn ; Lq ([0, 1]) ⊕ q ) n≥0

p p

p

n≥0

q

p

⊕ Lp (Ω∞ ; Lq ([0, 1]) ⊕ q )). p q It was also noted in [LR1] that (for fixed p, q) the class of ALp Lq -spaces is closed under ultraproducts. Lp {L∞ }-spaces We need in section 4 an appropriate version of the spaces Lp (L∞ ), 1 ≤ p < ∞. As before let (Ω1 , Σ1 , µ1 ) and (Ω2 , Σ2 , µ2 ) be two decomposable measure spaces. The space Lp (Ω1 , Σ1 , µ1 ) is order embedded into L0 (Ω1 × Ω2 , Σ1 ⊗ Σ2 , µ1 ⊗ µ2 ) by the map ϕ → ϕ ⊗ 1Ω2 . Then define the space Lp {Ω1 , Σ1 , µ1 ); L∞ (Ω2 , Σ2 , µ2 )} (in short Lp {L∞ }) as the K¨othe function space generated by Lp (Ω1 , Σ1 , µ1 ) in L0 (Ω1 × Ω2 , Σ1 ⊗ Σ2 , µ1 ⊗ µ2 ): f ∈ L0 (Ω1 × Ω2 , Σ1 ⊗ Σ2 ) belongs to Lp {L∞ } iff |f | is majorized by some function of Lp (Ω1 , Σ1 , µ1 ), and f Lp {L∞ } = inf{ϕLp | ϕ ∈ Lp (Ω1 , Σ1 , µ1 ), |f | ≤ ϕ}.  If we set: N∞ (f ) = {ϕ ∈ Lp (Ω1 , Σ1 , µ1 ) | ϕ ≥ |f |} then ||f ||Lp {L∞ } = ||N∞ (f )||p . (If the measure space (Ω2 , Σ2 , µ2 ) is σ-finite it can be shown that N∞ (f )(ω1 ) = Ess sup |f (ω1 , ω2 )| for a.e. ω1 ). ω2 ∈Ω2

This space contains isometrically as a sublattice the Banach lattice Lp (L∞ ) = Lp (Ω1 , Σ1 , µ1 ; L∞ (Ω2 , Σ2 , µ2 )) (of Bochner p-integrable L∞ -valued functions); it differs from it as soon as the space L∞ (Ω2 , Σ2 , µ2 ) is infinite-dimensional. If p is the conjugate exponent of p, the space Lp {L∞ } can be shown to be equal to the K¨ othe dual space Lp (L1 ) (with identical norm), see [HLR], Proposition 8.10 (or [BP], Corollary of Theorem 1 when the measure spaces are σ-finite) (if p > 1 it is the ordinary dual space Lp (L1 )∗ ). For that reason, this space is denoted by Lp∗ (L∞ ) in the monograph [HLR].

2. The Range of a Positive Contractive Projection in BLp Lq -Banach Lattices. The main result in this section (Theorem 2.8) is that the range of any contractive positive projection on a BLp Lq -Banach lattice is isometrically lattice isomorphic

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to a BLp Lq -Banach lattice. From this it follows (Theorem 2.9) that the class of BLp Lq -Banach lattices is closed under ultraroots and hence (Corollary 2.10) this class is axiomatizable by positive bounded sentences from the language of Banach lattices [HI]. First, we recall some facts. A Banach lattice X is strictly monotone (or has strictly monotone norm) if whenever x, y are elements of X with 0 ≤ x ≤ y, x = y one has x < y. Fact 2.1. In every strictly monotone (real or complex) Banach lattice X, the range of any positive contractive projection is a closed sublattice of X. Proof. Let P be a positive contraction on X and R(P ) be its range. If x ∈ R(P ) we have 0 ≤ |x| = |P x| ≤ P |x| and P ( |x|)  ≤  |x| , hence |x| = P ( |x|) by strict monotonicity, so |x| ∈ R(P ). Consequently R(P ) is a sublattice of X.
If X is a K¨ othe function space over a measure space (Ω, Σ, µ), f0 is a Σ-measurable positive function, and Σ0 is a sub-σ-algebra of Σ, we denote by Xf0 (Σ0 ) the space of classes of Σ0 -measurable functions h such that the support Sh of h is included in the support Sf0 of f0 and f0 h ∈ X. Then f0 · Xf0 (Σ0 ) := {f0 h | h ∈ Xf0 (Σ0 )} is a closed sublattice of X on which L∞ (Σ0 ) acts by ordinary multiplication. In what follows, we assume that all measure spaces (Ω, Σ, µ) are complete and decomposable. Fact 2.2. If X is an order-continuous K¨ othe function space over a measure space (Ω, Σ, µ), then for every closed sublattice Y of X there exists a sub-σ-algebra ΣY of Σ and a positive Σ-measurable function f0 such that Y = f0 · Xf0 (ΣY ). Proof. Let Σ0Y be the set of the supports of elements of Y . Then Σ0Y isa subσ-ring of Σ; indeed, we have Sf ∪Sg = S|f |∨|g| , Sf \Sg = S(|f |−|f |∧|g|)+ and Sfn = S(Σn αn |fn |) , where αn = 2−n fn −1 . Let ΣY be the σ-algebra generated by Σ0Y (which consists of the elements of Σ0Y and their complements). Let (fα ) α∈A be a maximal system of pairwise disjoint positive elements of fα be its l.u.b. in L0 and S0 = Sf0 its support. Y . Let f0 = α∈A

If f ∈ Y , then Af := {α ∈ A | µ(Sf ∩ Sfα ) > 0} is  at most countable (by Sα belongs to ΣY , it the order-continuity of X). Then Sf ⊆ S0 . (Since Sf \ α∈Af

is the support of an element g ∈ Y which is disjoint from all fα , α ∈ A; hence it is zero by the maximality of (fα )α∈A .) For every f ∈ Y+ and λ ∈ IR + , we have fα )+ ∈ Y (f −λf0 )+ ∈ Y ; indeed, for every finite subset F of A we have (f −λ α∈F  fα )+ , where F ranges over the finite and the downward directed net (f − λ α∈F

subsets of A, is order convergent, hence norm-convergent, to (f − λf0 )+ . Then Supp ( ff0 −λ1)+ = Supp (f −λf0 )+ ∈ Σ0Y , for every λ ≥ 0, while Supp ( ff0 −λ1)+ = Ω ∈ ΣY for every λ < 0. Hence f /f0 is ΣY -measurable and f ∈ f0 · Xf0 (ΣY ). By linearity this remains true for every f ∈ Y .

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Conversely, if f ∈ f0 · Xf0 (ΣY ) we have f = f0 h where h is ΣY -measurable with Sh ⊆ Sf0 . By linearity we may assume that f ≥ 0. Then h ≥ 0 is the l.u.b. of a net (hi ) of ΣY -measurable simple functions. To prove f ∈ Y it is sufficient to show that every f0 hi belongs to Y (since the net (f0 hi ) converges in the norm of X to f0 h = f by order-continuity of X). By linearity we simply have to prove that f0 1A belongs to Y whenever A ∈ ΣY . Again by order continuity it is sufficient to prove that fα 1A ∈ Y , for all α ∈ A and A ∈ ΣY . Note that we may assume that A ⊂ Sfα (if necessary, replacing A by A ∩ Sfα ). By the definition of ΣY , we know that A or its complement Ac belongs to Σ0Y . If Ac ∈ Σ0Y then A = Sfα \Ac belongs to Σ0Y too. Hence A ∈ Σ0Y in both cases; i.e., A = Sg for some g ∈ Y+ , so
fα 1A = sup fα ∧ ng, which belongs to Y by the order continuity of X. n≥1

Recall that if X is a K¨ othe function space and r ≥ 1, the r-convexification of X is the space X (r) of measurable functions f such that |f |r ∈ X, equipped with 1/r the norm f X (r) =  |f |r X (see [LT], p.53). Then X (r) is a Banach lattice for the natural order. For example if X = Lp (Lq ), then X (r) = Lpr (Lqr ). Recall also that a Banach space X is said to be smooth if for every nonzero x ∈ X the set of norming functionals for x, i. e. of x∗ ∈ X ∗ such that x∗  = 1 and x∗ , x = x, is reduced to an unique element denoted by Jx. (J is the “duality map” for X). If X is an order continuous, smooth and strictly monotone K¨ othe function space then for every nonzero f ∈ X, f and Jf have the same support. To prepare for the statement of the next Lemma, we consider the situation where the norm of X is strictly monotone, P is a positive contractive projection on X, and Y = R(P ) is the range of P . Denote by Σ0P , resp. ΣP the σ-ring, resp. σ-algebra generated by the supports of the elements of R(P ). Let ΣP be the order completion of ΣP , i.e., the closure of ΣP under the operations of l.u.b. and g.l.b. of infinite families. Note that, in particular, the support SP of R(P ) (which is the least element of Σ µ-almost containing all the Sf , f ∈ R(P )) belongs to ΣP , since it is the l.u.b. of the set {Sf | f ∈ R(P )}. In fact, ΣP consists of the sets S that are the l.u.b. of some subfamily of Σ0P ∪ {SPc }, where SPc = Ω\SP . It is easy to see 0 that a set A ∈ Σ belongs to ΣP iff its intersection with every  set B ∈ ΣP belongs 0 c c to ΣP while A ∩ SP =µ SP or ∅ (using the equality A = {A ∩ B | B ∈ Σ0P ∪ {SPc }} and the fact that for every B ∈ Σ0P the sigma-ring B ∧ Σ0P is closed under arbitrary l.u.b.’s, since B is σ-finite). Consequently a Σ-measurable function f is ΣP -measurable iff its restriction to any element of Σ0P is ΣP -measurable and its restriction to SPc is constant. In particular, if f has σ-finite support included in Sp , it is ΣP -measurable iff it is ΣP -measurable. Lemma 2.3. Assume that X is an order-continuous K¨ othe function space with strictly monotone norm and that some convexification X (r) is smooth. Let P be a positive contractive projection on X with range R(P ). Then there exists a positive Σ-measurable function f0 such that 1SP · R(P ∗ ) = f0 · Xf∗ (ΣP ). 0

Proof. Note that since X is order-continuous, its dual is a K¨ othe function space over the same measure space, namely the associated function space (or K¨othe dual)

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X  = {f  ∈ L0 (Ω, Σ, µ) | f  g ∈ L1 (Ω, Σ, µ) for every g ∈ X}. If X is reflexive, smooth and strictly convex (hence strictly monotone) (for example if X is a Lp (Lq )space with 1 < p, q < ∞), the proof is very simple. In this case X ∗ is the K¨othe dual of X and it is also order continuous, smooth and strictly convex. Since P ∗ is a positive contraction on X ∗ , its range R(P ∗ ) is by Fact 2.1 a closed sublattice of X ∗ , which by Fact 2.2 takes the form R(P ∗ ) = f0 · Xf∗ (ΣP ∗ ) for some positive 0 Σ-measurable function f0 . Let J: X \ {0} → X ∗ be the duality map, which is well defined since X is smooth. Then J(R(P )) ⊂ R(P ∗ ) (see [C]). Since X ∗ is smooth, we have also a duality map J∗ : X ∗ \{0} → X ∗∗ = X, and it turns out that J∗ : SX ∗ → SX is the inverse of J : SX → SX ∗ . So J induces a bijection between the unit spheres of R(P ) and R(P ∗ ). For every non-zero h ∈ X we have Supp Jh = Supp h; hence ΣP ∗ = ΣP . Finally since X ∗ is order-continuous, each of its elements has σ-finite support and so Xf∗ (ΣP ) = Xf∗ (ΣP ). 0 0 We now treat the general case (which will cover the case where X is a Lp (Lq )space with p = 1 or q = 1). We remark first that for every A ∈ ΣP with A ⊂ SP and every f ∈ X we have: P (1A f ) = 1A P (1SP f ).

(2.1)

Suppose first that A = Supp g for some positive g ∈ R(P ). Since P is positive it maps the closed order ideal generated by g (which is also the band generated by g, since X is order continuous) into itself. Hence P (1A f ) = 1A P (1A f ). By order continuity of X this last equation remains valid for an arbitrary ΣP measurable subset A of SP . If B = SP \A we have then P (1B f ) = 1B P (1B f ), so 1A P (1B f ) = 0 and eq. (2.1) follows. By duality we deduce that 1A P ∗ f  = 1SP P ∗ (1A f  ) 

(2.2)

∗

for every A ∈ ΣP with A ⊂ SP and every f ∈ X . For every 0 = f ∈ X, there exists a norming functional Jf ∈ X ∗ such that Supp Jf ⊂ Supp f ; in fact, Supp Jf = Supp f (due to the strict monotonicity of the norm in X); moreover any other norming functional f  for f coincides with Jf on the support of f (use the fact that f  (|f |/f X )1−1/r is a norming functional for |f |1/r sgn f ∈ X (r) )). We claim that we have: 0 = h ∈ R(P ) =⇒ Jh = 1Sh P ∗ Jh ∈ 1SP · R(P ∗ ). ∗

(2.3)

∗

Indeed, P Jh, h = Jh, P h = Jh, h = h and P Jh ≤ Jh = 1 imply that P ∗ Jh is a norming functional for h, so 1Sh P ∗ Jh = Jh. But by (2.2), 1Sh P ∗ Jh = 1SP P ∗ (1Sh Jh) = 1SP P ∗ Jh. Now let (fα )α∈Abe a maximal  family of positive, pairwise disjoint elements fα , f0 = α∈A Jfα . If h ∈ X with Supp h ⊂ Supp fα of R(P ), and f0 = α∈A

we have Supp P h ⊂ Supp fα by (2.1), hence we can define Qα : Xfα → Xfα by (Qα ϕ)fα = P (ϕfα ). Let us denote by να the measure (fα Jfα ) · µ, which is positive and bounded (of mass fα X ). Note that L∞ (να ) ⊂ Xfα ⊂ L1 (να ) since

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  for every ϕ ∈ Xfα , |ϕ| dνα = (|ϕ|fα )Jfα dµ ≤ ϕ.fα X . Similarly L∞ (να ) ⊂ ∗ ⊂ L1 (να ). Let us identify Xfα with 1Sα · X by defining ϕ = ϕfα X XJf α ∗ for each ϕ ∈ Xfα and similarly we identify XJf with 1Sα · X ∗ . Note that if α ∗ ϕ ∈ Xα ,ψ ∈ XJfα , then:   ϕψ dνα = (ϕfα )(ψJfα ) dµ ∗ ∼ so XJf = (Xfα )∗ for the pairing associated with integration with respect to να . α ∗ Moreover for every ϕ ∈ Xfα , ψ ∈ XJf we have: α    ∗ ϕQα ψ dνα = (Qα ϕ)ψ dνα = P (ϕfα )ψJfα dµ   1S P ∗ (ψJfα ) ∗ = ϕfα P (ψJfα ) dµ = ϕ α dνα Jfα

and hence: 1Sα P ∗ (ψJfα ) = (Q∗α ψ)Jfα .

(2.4)

Note that 1Sα is preserved by Qα and Q∗α since P fα = fα and 1Sα P ∗ Jfα = o (see [DHP], Proposition 4.7), the Jfα (by 2.3). By a classical argument of Andˆ contractive projection Qα extends by continuity to a conditional expectation Eα on L1 (Sfα , να ), which is necessarily relative to the trace Σα of ΣP onto Sfα . Then the restriction of Qα to L∞ is bounded L∞ → L∞ (and coincides still with the conditional expectation Eα ), and consequently the adjoint operator Q∗α extends to an operator L1 → L1 which is nothing but the same conditional expectation Eα . ∗ ) ⊂ L1 (Sα , Σα , να ) contains only Σα -measurable functions; that is, Thus Q∗α (XJf α ∗ ∗ ∗ Qα (XJfα ) ⊂ XJf (Σα ). α Consequently from (2.4) we obtain: ∗ 1Sα P ∗ (ψJfα ) ∈ Jfα · XJf (Σα ) α

and for every f  ∈ X ∗ we have by (2.2) (setting 1Sα f  = ψα Jfα for every α): 1SP P ∗ (f  ) = 1Sα P ∗ f  = 1Sα P ∗ (1Sα f  ) = (Q∗α ψα )Jfα α

=

f0

α

α

1Sα Q∗α ψα

=

α

f0 ϕ

 where ϕ = α 1Sα Q∗α ψα is ΣP -measurable as (µ-almost) disjoint sum of ΣP -measurable elements. Consequently 1SP P ∗ (f  ) ∈ f0 · Xf∗ (ΣP ). 0 Conversely if A ∈ ΣP then for every α, 1A Jfα ∈ 1SP R(P ∗ ) by (2.3) and (2.2). If moreover 1A f0 ∈ X ∗ it follows that 1A f0 ∈ 1SP · R(P ∗ ). (Note that in X ∗ , every downwards directed net (xα ) whose g.l.b. is 0 converges weak* to zero, since for every x ∈ X+ , the family (xxα ) is a downwards directed net in L1 (µ) with g.l.b. 0, hence converges to 0 in L1 -norm. Thus, since R(P ∗ ) is weak*-closed in X ∗ , any element of X ∗ which is the l.u.b. of an upwards directed net in R(P )∗ belongs,

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in fact, to R(P )∗ ). A similar weak*-approximation argument by simple functions shows that f0 · Xf∗ (ΣP ) ⊂ 1SP · R(P ∗ ).
0

Now we consider the situation where X is an Lp (Lq )-space. Let (Ω1 , Σ1 , µ1 ) and (Ω2 , Σ2 , µ2 ) be two decomposable measure spaces and X = Lp ((Ω1 , Σ1 , µ1 ); Lq (Ω2 , Σ2 , µ2 )); this space is considered as a K¨othe function space over the mea¯ 2 , µ1 ⊗ µ2 ). Then Σ1 and Σ2 are naturally identified sure space (Ω1 × Ω2 , Σ1 ⊗Σ ¯ 2 . Let Nq : X → Lp (Ω1 , Σ1 , µ1 )+ be with Boolean sub-algebras of Σ := Σ1 ⊗Σ the natural random norm defined in §1. If f ∈ X, we will denote the support of Nq (f ) by Ωf and call it the Σ1 -support of f ; Ωf is also the µ1 -least element of Σ1 containing the ordinary support Sf of f . Lemma 2.4. Suppose 1 ≤ p = q < ∞ and let P be a positive contractive projection in an Lp (Lq )-space X. Let R(P ) = f0 · Xf0 (ΣP ), where ΣP is the σ-algebra generated by the supports of elements of R(P ) as in Fact 2.2. Then, for any elements Nq (h) 1S is ΣP -measurable. f, h ∈ R(P ) with Sh ⊂ Sf , the function Nq (f ) f Proof. By Lemma 2.3 we have 1SP · R(P ∗ ) = f0 · Xf∗ (ΣP ), for some positive 0 ¯ 2 -measurable function f0 . Let J: X → X ∗ be the duality map, (defined as Σ1 ⊗Σ in the proof of Lemma 2.3 when p = 1 or q = 1). Then J(R(P )) ⊂ 1SP · R(P ∗ ). (See the proof of Lemma 2.3). Jh h and are well defined (with the usual Since Sh ⊂ Sf , the quotients f Jf 0 convention = 0) and ΣP -measurable. Since 0 |h|q−1 sgn h Jh = Nq (h)q−p hp−1 we have Jh = Jf so



Nq (h) Nq (f )



f  h

p−1 q−1 p−q

h h Nq (h)

sgn

f f Nq (f )

p−q 1Sh =

h f 

p−1

q−1 Jh

f

f sgn

Jf h h

Nq (h) 1S is ΣP -measurable too. Nq (f ) h Note that there exists a sequence (εn ) → 0 such that Supp (h + εn f ) = Supp (f ) for every n (since the set {ε ∈ R | Supp (h + εf ) = Supp (f )} is at most Nq (h + εn f ) 1Sf is ΣP -measurable, and passing to the limit, so countable). Then Nq (f ) Nq (h) is 1S .
Nq (f ) f is ΣP -measurable. Since p − q = 0 the function

Corollary 2.5. For every h ∈ R(P ), the set Ωh ∩ SP is ΣP -measurable.

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Proof. By the preceding, Ωh ∩ Sf belongs to ΣP for every f ∈ R(P ) with Sf ⊃
Sh . Lemma 2.6. For every f, h ∈ R(P ) with Ωh ⊂ Ωf , the function

Nq (h) 1Ω ∩S is Nq (f ) f P

ΣP -measurable. Nq (h) 1Ω ∩S Nq (f ) f g is ΣP -measurable. But by the preceding corollary, the set Ωf ∩ Sg belongs to ΣP ; i.e., it has the form Sg for some g  ∈ R(P ). We may assume w.l.o.g. that g  = g Nq (h) Nq (f ) 1S and 1S are and that Sg ⊃ Sf ∪ Sh . By Lemma 2.4 the functions Nq (g) g Nq (g) g Nq (h) ΣP -measurable; hence, so is their quotient 1S .
Nq (f ) g

Proof. It is sufficient to prove that for every g ∈ R(P ), the function

Now let ΩP be the Σ1 -support of R(P ), i.e., the supremum of all Ωh , h ∈ R(P ), which is also the g.l.b. of the family of sets in Σ1 which contain SP . Lemma 2.7. There exists ϕ0 ∈ L0 (Ω1 , Σ1 , µ1 )+ with support ΩP , such that for Nq (h) every h ∈ R(P ), the function 1SP is ΣP -measurable. ϕ0 Proof. We consider a maximal family (gi )i∈I of elements of R(P ) such that the Nq (gi ) are pairwise disjoint.  Ωgi = ΩP ; if not, there exists g ∈ R(P ) with Ωg ⊂µ Ω := We claim that i∈I  Ωgi ; that is, 1(Ω )c g = 0. Note that by Corollary 2.5, Ω ∩ SP belongs to ΣP .

i∈I

Then the set U := Sg\Ω is not µ-almost void and belongs to ΣP ; hence g  := 1U ·g is non-zero and belongs to R(P ), and Ωg ∩ Ω = ∅; this contradicts the maximality of the family (gi )i∈I .  Nq (gi ) (taking the supremum in L0 (Ω1 , Σ1 , µ1 )). By We now set ϕ0 = i∈I

the preceding we have Ωϕ0 = ΩP ; moreover for every h ∈ R(P ) and i ∈ I, Nq (h) Nq (hi ) 1Ω ∩S (where hi = 1Ωgi h); by Lemma 2.6 this ratio 1Ωgi ∩SP = ϕ0 Nq (gi ) gi P of functions is ΣP -measurable; hence the supremum of these ratios is also measurNq (h) able, and it is 1SP .
ϕ0 Theorem 2.8. Let 1 ≤ p, q < ∞. Then the range of any contractive positive projection in a BLp Lq -Banach lattice is isometrically lattice isomorphic to a BLp Lq Banach lattice. Proof. Since a BLp Lq -Banach lattice is (isometrically lattice isomorphic to) a band in a concrete Lp (Lq )-Banach lattice, and such a band is itself positively contractively complemented (by the associated band projection) we may restrict to the case of a contractive positive projection P on a concrete Lp (Lq )-Banach lattice X.

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By Fact 2.2, the range R(P ) of P is a closed sublattice of X of the form f0 · Xf0 (ΣP ), where ΣP is the σ-algebra generated by the supports of the elements ¯ 2 ) with support SP (the support of R(P ) and f0 is a positive element of L0 (Σ1 ⊗Σ Nq (h) of R(P )). By Lemma 2.7, for every element h ∈ R(P ) the function 1SP is ϕ0 ΣP -measurable, where ϕ0 is some positive element of L0 (Σ1 ) with support ΩP (the Σ1 -support of R(P )). Let Σ1,P be the set of elements A of Σ1 which are subsets of ΩP and for which A ∩ SP belongs to ΣP : this is a complete Boolean algebra with maximal element ΩP . Let ν be the restriction of the measure ϕp0 · µ1 to Σ1,P . Note that the measure ν is semi-finite since ΩP is a supremum of sets Ωgi , gi ∈ R(P ) which belong to Σ1,P by Corollary 2.5 and for which   ν(Ωgi ) = ϕp0 dµ1 = Nq (gi )p dµ1 = gi pX < ∞. Ωgi

The space R(P ) is an L∞ (Σ1,P )-module. Indeed, if h ∈ R(P ) and A ∈ Σ1,P then 1A h = 1A∩SP h belongs to R(P ), because A ∩ SP ∈ ΣP and R(P ) is clearly an Nq (h) L∞ (ΣP )-module. For every h ∈ R(P ), the function Nq,P (h) = is Σ1,P -meaϕ0 surable (by the definition of Σ1,P ) and moreover   p Nq,P (h) dν = Nq (h)p dµ1 = hpX . The map Nq,P : R(P ) → Lp (ΩP , Σ1,P , ν)+ is thus an Lp (ν)-valued q-random norm on the Banach lattice R(P ). In fact, since Σ1,P ⊂ Σ1 , we clearly have: Nq,P (ϕh) = |ϕ|Nq,P (h) for every h ∈ R(P ), ϕ ∈ L∞ (Ωp , Σ1,P , ν) and the other axioms of q-random norm are clearly inherited from the original Nq function. Consequently R(P ) is an abstract Lp (Lq )-space; its underlying Banach
lattice is therefore a BLp Lq -Banach lattice. Theorem 2.9. Let 1 ≤ p, q < ∞. In the category of Banach lattices, the class of BLp Lq -Banach lattices is closed under ultraroots. In other words, if a Banach lattice X has an ultrapower XU which is isometrically lattice isomorphic to a BLp Lq Banach lattice, then X itself is isometrically lattice isomorphic to a BLp Lq -Banach lattice. Proof. (a) Assume first that p, q > 1: then XU is reflexive, and consequently so is X. Then it is well known that the canonical image of X in XU (by the diagonal embedding ıU ) is 1-complemented by a positive projection, namely: P U : XU → X, [xi ]U → w - lim xi . i,U

(where w- lim denotes the weak limit). By Theorem 2.8, X is isometrically lattice isomorphic to a BLp Lq -lattice.

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(b) In the nonreflexive case (when p or q equals 1) note that we can still define a contractive map from XU into the bidual X ∗∗ of X, using now weak* limits: P U : XU → X ∗∗ , [xi ]U → w∗ - lim xi . i,U

As XU ∼ = Lp (Lq ) does not contains c0 isomorphically as a subspace, its subspace X does not either; since X is a Banach lattice it results from [LT], Theorem 1.c.4 that X is a projection band in its bidual X ∗∗ . Let πX : X ∗∗ → X be the corresponding band projection, we have then πX P U ıU = idX , which means that ıU πX P U is a (positive, contractive) projection from XU onto its sublattice ıU (X) ∼ = X. Then we may again finish the proof by applying Theorem 2.8.
Corollary 2.10. The class of BLp Lq -Banach lattices is axiomatizable in the language of Banach lattices by positive bounded sentences. Proof. Since this class is closed under ultraproducts ([LR1]) and ultraroots (Theorem 2.9) this is a consequence of [HI], Proposition 13.6.


3. Paving by Finite Dimensional BLp Lq -Banach Lattices. In this section we give another, more intrinsic characterization of BLp Lq -Banach lattices (Proposition 3.6). It is expressed in terms of pavings by certain finite dimensional Banach lattices, and is completely analogous to the characterization of Lp -Banach spaces as the Banach spaces that are paved by the spaces (np | n ≥ 1) almost isometrically (i.e., as the (Lp )1 -Banach spaces) (see e.g. [L], p.167). At the end of the section we quantitatively sharpen our paving characterization of BLp Lq Banach lattices and are thereby able to give an explicit set of axioms for this class. Definition 3.1. A Banach lattice X is a (Lp Lq )λ -lattice if for every ε > 0 and every finite system (x1 , . . . , xn ) of positive disjoint elements of X there exists a finite dimensional sublattice F of X which is (λ+ε)-lattice isomorphic to a finite dimensional BLp Lq -Banach lattice and contains elements x1 , . . . , xn such that xj − xj  ≤ ε for all j = 1, . . . , n. Remark. The finite dimensional BLp Lq -Banach lattices are the ones isometrically lattice isomorphic to a p-direct sum of a finite sequence of finite dimensional Lq Banach spaces, namely to nq 1 ⊕p · · · ⊕p nq k for some finite sequence n1 , . . . , nk of integers ≥ 1. Lemma 3.2. Assume that X is a (Lp Lq )λ -lattice. Then for every ε > 0 and every finite dimensional sublattice E of X there exists another finite dimensional sublattice F of X and a vector lattice homomorphism T from E into F such that F is (λ + ε)-lattice isomorphic to a finite dimensional BLp Lq -Banach lattice and T x − x ≤ εx for every x ∈ E.

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Proof. Let (x1 , . . . , xn ) be a system of norm-one atoms of E and δ > 0. Let F be a finite dimensional sublattice of X which is (λ + δ)-lattice isomorphic to a finite dimensional BLp Lq -Banach lattice and contains elements x1 , . . . , xn such that xj − xj  ≤ δ for all j = 1, . . . , n. First we show that we may suppose the xj ’s to be positive and pairwise disjoint. For the first point note that for every j, |xj | ∈ F and ||xj | − xj | ≤ |xj − xj |. For the second point note that we have |xj | ∧ |xk | = |xj | ∧ |xk | − xj ∧ xk  ≤ xj − xj  + xk − xk  ≤ 2δ. Setting

  yj = |xj | − |xj | ∧  |xk | k =j

we obtain positive disjoint elements of F such that yj − xj  ≤ 2nδ, j = 1, . . . , n. Now let T : E  → F be the linear operator defined by T xj = yj , j = 1, . . . , n. Then if x ∈ E, x = i αi xi we have T x − x ≤ |αi |T xi − xi  ≤ n(2n + 1)δx i

(since |αi | ≤ x, i = 1, . . . , n). Given any ε > 0, applying this argument with δ chosen so that n(2n + 1)δ = ε completes the proof.
Proposition 3.3. Every (Lp Lq )λ -lattice X is isometrically lattice isomorphic to a positively and contractively complemented sublattice of a Banach lattice which is λ-lattice isomorphic to a BLp Lq -Banach lattice. Proof. It results from the preceding lemma that X satisfies a lower r-estimate for some r < ∞ (in fact, r = max(p, q)): for every finite system x1 , . . . , xn of disjoint elements,    r   xi  ≥ λ−r xi r .    i

i

Then X does not contain c0 as a sublattice, so X is order continuous (in fact, a “KB-space”; see [MN], Theorem 2.4.12), and is therefore representable as a K¨ othe function space ([LT], Theorem 1.b.14). Consequently X is the closure of the union X0 of some net (Ei ) of its finite dimensional sublattices (e.g., generated by simple functions). For each i and each ε > 0, let Ti,ε be a lattice homomorphism from Ei into another finite dimensional sublattice Fi,ε of X, itself (λ+ε)-lattice isomorphic to a BLp Lq -Banach lattice, such that Ti,ε x − x ≤ εx for every x ∈ Ei . Let U be an ultrafilter on I × (0, ∞) containing all of the final  sections Si0 ,ε0 = {(i, ε) | i > i0 , ε < ε0 }. Then the ultraproduct map T = U Ti    =  is an isometric lattice embedding from E U Ei into F = U Fi,ε , and this last lattice is λ-lattice isomorphic to an ultraproduct of BLp Lq -Banach lattices,

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 hence to a BLp Lq -Banach lattice. Moreover X0 = i Ei embeds naturally (as a  =  Ei by the diagonal embedding D0 ; this is defined by: normed lattice) into E U D0 (x) = (Di (x))U where Di (x) = x if x ∈ Ei , Di (x) = 0 if not. The embedding D0 extends in a unique way (by density and uniform continuity) to an embedding  Then TD is the desired embedding. If X is reflexive we may define D of X into E. a contractive positive linear map P : F → X by P ([xi ]U ]) = w − lim xi . Since (i,ε)U

for every x ∈ X0 we have Ti,ε x − x → 0 we have P TDx = x for every x ∈ X0 , (i,ε)U

hence for every x ∈ X by density and continuity, so P is a projection. In the case where X is not reflexive we remark that X is nevertheless a band in its bidual X ∗∗ [see the proof of Theorem 2.9] and may consider the corresponding band projection πX ; now let S be the map F → X ∗∗ , S([xi ]U ) = w∗ − lim xi . Then P = πX S is (i,ε)U

the desired projection.




Remark 3.4. Since X is representable as an order continuous K¨ othe function space it is not hard to see that the following strengthening of Definition 3.1 holds true: A Banach lattice X is a (Lp Lq )λ -lattice if for every ε > 0 and every finite system (x1 , . . . , xn ) of elements of X there exists a finite dimensional sublattice F of X which is (λ + ε)-lattice isomorphic to a finite dimensional BLp Lq lattice and contains elements x1 , . . . , xn such that xj − xj  ≤ ε for all j = 1, . . . , n. Lemma 3.5. Every BLp Lq -Banach lattice is a (Lp Lq )1 -lattice. Proof. It is clearly sufficient to prove the assertion for separable BLp Lq -Banach lattices. It will be done if we exhibit in such lattices an increasing sequence (Hn ) of sublattices such that each of them is (1 + εn )-lattice isomorphic (or better: isometric) to a finite-dimensional BLp Lq -Banach lattice and such that the union is dense. A separable BLp Lq -Banach lattice has a concrete representation as a band in a separable Lp (Lq )-space. We may suppose that in this representation the Lq -space is concretely represented as Lq [0, 1] ⊕q q . Our BLp Lq -space X is now represented as X = 1S · Lp (Ω, Σ, µ; Lq [0, 1] ⊕q q ), where S is a measurable subset in the product measure space Ω × ([0, 1] ∪ IN). The sub-band X0 = 1S · Lp (Lq [0, 1]) can be also described as Xc = 1Sc · Lp (Lq [0, 1]), where Sc is the trace of S on Ω × [0, 1]. This band in turn is isometrically lattice isomorphic to Y = Lp (Ac ; Lq [0, 1]) where A is some measurable subset of Ω, by some random-norm preserving lattice isomorphism. (See [HLR], Theorem 8.7 where a more general result is stated as consequence of Maharam’s decomposition theorem). Let Sd = S \ Sc and Xd = 1Sd · Lp (Lq [0, 1] ⊕q q ) ∼ = 1U .Lp (q ). We have then X ∼ = Y ⊕q Xd , with the random norm on this direct sum of ALp Lq -spaces being given by the q-addition rule NX (f ⊕ g) = NY (f )q + NXd (g)q . We may write Sd = j Aj × {ej } where the ej are the atoms of the lattice q , and Aj is some measurable subset of Ω. Now let (En ) be an increasing sequence of finite dimensional sublattices of Lp [0, 1] (hence lattice isometric to dpn spaces) with dense union; we may assume that 1Ac ∈ E1 and 1Aj ∈ Ej for all j. Let similarly

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(Fn ) be an increasing sequence of finite n dimensional sublattices of Lq [0, 1]. Set Gn = 1Ac · En ⊗ Fn and Hn = Gn ⊕ j=1 1Aj En ⊗ ej . Then (Hn ) is an increasing sequence of finite dimensional sublattices of Y ⊕ Xd , each of them being isometrically lattice isomorphic to a BLp Lq -space, and their union is dense in Y ⊕ Xd .
Proposition 3.6. Let 1 ≤ p, q < ∞. A Banach lattice is a (Lp Lq )1 -lattice if and only if it is isometrically lattice isomorphic to a BLp Lq -Banach lattice. Proof. We know by Lemma 3.5 that every BLp Lq -Banach lattice is a (Lp Lq )1 lattice so we only need to prove the converse inclusion. But by Proposition 3.3 every (Lp Lq )1 -lattice is isometrically lattice isomorphic to a positively contractively complemented sublattice of an BLp Lq -Banach lattice; hence by Theorem
2.8 it is itself isometrically lattice isomorphic to an BLp Lq -Banach lattice. We now give a quantitative version of Lemma 3.5. Proposition 3.7. For every integer n, for every real ε > 0, for every BLp Lq -Banach lattice X and for every system S = {x1 , . . . , xn } of disjoint positive elements in X there exist a sublattice F of X of dimension d(F ) ≤ d(n, ε) which is isometrically lattice isomorphic to a BLp Lq -Banach lattice and a map f : S → F such n + that f (xi ) − xi  ≤ ε i=1 xi , i = 1, . . . , n. One can choose d(n, ε) = n2n[1/ε] + (where [1/ε] denotes the least integer greater than 1/ε). Proof. We may suppose that ε = 1/N and that X is a band in some Lp (Ω, Σ, µ; Lq )space. For every i = 1, . . . , n and k = 0, . . . , N − 1, let   k k+1 ϕ Ai,k = ω ∈ Ω | ϕ < N (xi ) ≤ N N n N k where we have set ϕ = i=1 N (xi ). Let ϕi = k=1 N 1Ai,k ϕ then 0 ≤ N (xi ) − ϕi ≤ εϕ for every i = 1, . . . , n. Then setting yi = we have

ϕi xi N (xi )



ϕi

− 1

N (xi ) = |ϕi − N (xi )| ≤ εϕ N (xi − yi ) = N (xi )

and consequently xi − yi  ≤ ε

n

xi .

i=1

Note that the nN sets Ai,k , where 1 ≤ i ≤ n and 0 ≤ k < N , generate a ring of measurable subsets, with no more than 2nN minimal elements. Hence there is a system (B )=1,...,r of disjoint measurable subsets of Ω, with r ≤ 2nN such that ϕ |xi | every Ai,k is a finite union of a sub-collection of the sets B . Set zi, = 1B N (x i)

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(with the convention 0/0 = 0); the elements yi are linear combinations of the elements zi, . The latter are positive, disjoint and still belong to the band generated by the xi ’s. Note that for every , all the elements zi, which are not zero have the same random norm, namely 1B ϕ, and if  =  then N (zi, ) is disjoint of N (zi, ). Consequently the elements zi, generate a sublattice isometrically lattice
isomorphic to some BLp Lq -Banach lattice of dimension at most n2nN . Axioms for BLp Lq -Banach lattices. Now we describe briefly how to obtain from Proposition 3.6 a set of axioms characterizing the class of BLp Lq -Banach lattices. Note that a mathematical object like a finite dimensional sublattice F of X cannot appear in formulas written in the language of Banach lattices, which involves only scalars, elements of the space and Banach lattice operations; this has to be circumvented by considering in place of F a finite generating set y of atoms of F . First let us see how to express that y generates a finite dimensional BLp Lq -space. Such a space has di the form (⊕m i=1 q )p , so we have to give a double indexing of the tuple y, setting y = (yij ) i=1,...,m . Let d = (d1 , . . . , dm ). Then the formula j=1,...,di

 (y) : ∀(λij ), ψm,d,N

m i=1

 

di

q/p |λij |p 

1/p

j=1

   m di     ≤  λij yij    i=1 j=1   q/p  di m 1  ≤ 1+ |λij |p  1/p N i=1 j=1

expresses that the linear span of y is (1 + 1/N )-linearly isomorphic to the space di (⊕m i=1 q )p . The formula  ψm,d (y) : |yij | ∧ |yi j  | = 0 (i,j) =(i ,j  )

expresses that the elements of y are pairwise disjoint, and the formula:  (y) : ψm,d

di m

|yij − |yij | | = 0

i=1 j=1

expresses that these elements are positive. Hence the conjunction ψm,d,N (y) of these three formulas expresses that y generates a sublattice of X which is (1+1/N )di lattice isomorphic to the space (⊕m i=1 q )p . Given the system x = (x1 , . . . , xn ) in X, let ϕn,m,d,N (¯ x) be the formula  di m    1 ∃λ xi − λij yij  ≤ ∃y ψm,d,N (y) ∧ N i=1,...,n i=1 j=1 and let φn,N (x) be the disjunction of the formulas ϕn,m,d,N (¯ x) when m, d vary m subject to the condition i=1 di ≤ d(n, 1/N ). Then φn,N (x) expresses that there

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is a sublattice F of X which is (1 + 1/N )-lattice isomorphic to a BLp Lq -normed latttice of dimension less than d(n, 1/N ) and such the distance to F of all members of x is less than ε. The desired set of axioms is then (An,N )n≥1,N ≥1 where An,N is the sentence:  (An,N ) : ∀x1 . . . ∀xn |xi | ∧ |xj | = 0 =⇒ φn,N (x) . i,j=1,...,n i=j

The translation of these axioms into the language of positive bounded sentences is left to the reader acquainted with [HI].

4. An Isometric Characterization of the Sublattices of the Lattices Lp (Lq ) Recall that in any Banach lattice X one can define for every q > 0 the operation X × X → X, (x, y) → (|x|q + |y|q )1/q by Krivine’s functional calculus. This expression is a norm limit of a sequence of lattice terms, i.e., of algebraic expressions hn (x, y) involving only the vector space and lattice operations and a finite number of scalars (see [LT], Theorem 1.d.1). In the case q ≥ 1, these lattice terms can be given as follows. Let (αi , βi ) be a dense subset in the set   {(α, β) ∈ IR2+ | αq + β q = 1}, where q  is the exponent conjugate to p. Then set hn (x, y) =

n 

(αi |x| + βi |y|).

i=1

Note that when X is a lattice of measurable functions (with its natural order) then the operation (|x|q + |y|q )1/q has the usual meaning. Theorem 4.1. Let 1 ≤ p < q < ∞. A Banach lattice X is (isometrically lattice isomorphic to) a sublattice of a space Lp (Lq ) iff X satisfies the following condition (where 1 ≤ n ∈ IN, x1 , . . . , xn ∈ X and λ1 , . . . , λn ∈ IR): ∀n ∀x1 . . . ∀xn ∀λ1 . . . ∀λn =⇒



 n

λi = 0

i=1

 1 λi λj (|xi |q + |xj |q ) q pX ≤ 0 .

(4.1)

1≤i,j≤n

Condition (4.1) means that the pth power of the norm is a negative definite function (in the sense of [BCR]) on the semi-group (X+ , +q ) where X+ is the positive cone of X and +q is the operation (x, y) → (xq + y q )1/q . The proof of Theorem 4.1 depends essentially on the following representation lemma for homogeneous negative definite functions; we give a proof in the Appendix:

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Lemma 4.2. Every negative definite IR+ -valued function f on the semi-group (IRd+ , +q ) which is homogeneous of degree p ∈ (0, q) (relatively to the ordinary scalar multiplication) has the form  p/q  q  ui xi  dσ(u) when x = (x1 , . . . , xd ) f (x) = B

1≤i≤d

where the ui ’s are non-negative real numbers, B is a compact base of the cone IRd+ and σ is a Radon measure on B. Another key ingredient is the following: Lemma 4.3. Let X be a Banach lattice. Assume that each finite dimensional sublattice of X is isometrically lattice isomorphic to a sublattice of an Lp (Lq )-Banach lattice (allowing the measure spaces to vary). Then there is an isometric lattice isomorphism of X itself into some Lp (Lq )-Banach lattice. Proof. If p ≤ q the Banach lattice X satisfies an exact q-lower estimate; i.e., for any family of disjoint vectors x1 , . . . , xN in X we have:   1/q N   N q   xi  ≥ xi   i=1

i=1

since this inequality is true for every family of disjoint elements of Lp (Lq ). Similarly if p ≥ q, then X satisfies an exact p-lower estimate. This implies that X is an order continuous Banach lattice (see e.g., [LT], Theorem 1.a.5 and Proposition 1.a.7), and can thus be represented as a K¨ othe function space over some measure space (S, Σ, m) ([LT], Theorem 1.b.14 and Proposition 1.a.9). It is then easy to find  a directed net (Xi )i∈I of finite dimensional sublattices such that X = i Xi (norm closure): take for I the set of finite systems Φi = {A1 , . . . , Adi ) of disjoint Σ-measurable subsets of finite m-measure and let Xi be the lattice of simple functions relative to Φi which belong to X. Let Si be an isometric linear lattice embedding of Xi into Lp (Ωi , Ai , µi ; dq i ), and U be an ultrafilter on I containing the final sections Fi = {j | Xj ⊇ Xi }. Let   S : i,U Xi → i,U Lp (Ω, Σ, µ; dq i ) be the ultraproduct of the embeddings Si : this  is an isometric lattice embedding of the ultraproduct X = i,U Xi into the ultra product L = i,U Lp (Ω, Σ, µ; dq i ), which is itself an ALp Lq -space. The diagonal  embedding D : i Xi → X defined by Dxi = [(Di,j xi )j ]U , (where Di,j (xi ) = xi if i ≤ j, = 0 if not), extends by completion to an isometric linear lattice embedding  X → L provides the desired isometric of X into X . The composition map SD, linear lattice embedding of X into an ALp Lq -space (which is itself a sublattice of
some Lp (Lq )-Banach lattice). Proof of Theorem 4.1. (a) Necessity.  Assume first that X = IR. Then for all reals t1 , . . . , tn ≥ 0 and λ1 , . . . , λn with i λi = 0,

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λi λj (ti +q tj )p =

1≤i,j≤n

Positivity

λi λj (tqi + tqj )p/q ≤ 0

1≤i,j≤n

since it is well known that the power functions x → xα are negative definite on the semigroup (IR+ , +) whenever 0 < α ≤ 1 (this can be derived for 0 < α < 1 from the trivial case α = 1 using e.g., [BCR], Corollary 3.2.10). Assume now that X = Lp (Lq ). Observe that f +q gLq = f Lq +q gLq for any f, g ≥ 0 in Lq . Hence:  λi λj  |xi | +q |xj | pX = λi λj Nq (|xi | +q |xi |)p dµ1 Ω1 1≤i,j≤n

1≤i,j≤n

 =



Ω1 1≤i,j≤n

λi λj [Nq (xi ) +q Nq (xj )]p dµ1

≤ 0 by the scalar case. (b) Sufficiency. By Lemma 4.3 it suffices to consider the case where X is a finite dimensional normed lattice verifying eq. (4.1). Let e1 , . . . , ed be the set of d λi ei atoms of X. Then the natural linear isomorphism IRd → X, (λ1 , . . . , λd ) → i=1

identifies the order of X with the natural partial order on IRd . So we identify X with IRd equipped with a lattice norm verifying (4.1). By Lemma 4.2, we find that  p/q  q  xp = ui |xi |q  dσ(u) B

1≤i≤d

for some B and σ. This formula gives an isometric lattice isomorphism from X into the space Lp (B, σ; dq ), mapping each ei onto Ui ⊗ fi , where fi is the i-th unit
basis vector of lqd and Ui is the i-th coordinate map u → ui . A similar but more complicated characterization holds true in the case p > q, provided the quotient p/q is not an integer: Theorem 4.4. Let 1 ≤ q ≤ p < ∞, and assume that p/q is not an integer. Let r = [ pq ] + 1 (where [t] denotes the integer part of the real number t) and let k be any integer ≥ r/2. A Banach lattice X is isometrically lattice isomorphic to a sublattice of a space Lp (Lq ) iff X satisfies the following condition (where 1 ≤ n ∈ IN, x1 , . . . , xn ∈ X and λ1 , . . . , λn ∈ IR):  n ∀n∀x1 . . . ∀xn ∀λ1 . . . ∀λn λi = 0 =⇒ (−1)r

1≤i1 ,...,i2k ≤n

i=1 1

λi1 . . . λi2k (|xi1 |q + · · · + |xi2k |q ) q pX ≥ 0

 (4.2)

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A key ingredient in the proof of Theorem 4.3 is the following theorem, due to J.-L. Krivine [K1]; we use only the finite dimensional version of this result. Fact 4.5. ([K1], Theorem 9, p. 38) Assume that α ≥ 1 is not an integer and let r = [α] + 1 and k be any integer greater than r/2. Let Γ be a finite dimensional closed convex cone in some linear space, with Γ ∩ (−Γ) = (0), and Φ : Γ → IR+ a continuous positively 1-homogeneous map. The following conditions are equivalent: (i) There exists a positively linear map T from Γ into the positive cone L+ α of some space Lα (Ω, Σ, µ) such that T x = Φ(x) for every x in Γ; (ii) The map (−1)r Φα is “2k-positive”, that is: ∀n ∈ IN ∀x1 , . . . , xn ∈ Γ ∀λ1 , . . . , λn ∈ IR

 n

λi = 0

i=1



r

=⇒ (−1)

λi1 . . . λi2k Φ(xi1 + · · · + xi2k )

α

 ≥ 0 . (4.3)

1≤i1 ,...,i2k ≤n

Remark. This result is stated in [K1] for Φ being the restriction of a norm to Γ; the subadditivity of Φ is however not used in the proof, and we will apply this result for a map Φ which is a priori not a norm. Proof of Theorem 4.4. (a) Necessity. Assume that X is a sublattice of Lp (Lq ). If x1 , . . . , xn ∈ X, let ϕi = Nq (xi )q ∈ L+ p/q . Then   1/q  1/q    p   p/q  p          q q  |xij | |xij | ϕij    = Nq   =   .     p/q j j  j  p

p ≥ 1 we have by Fact 4.5 (taking for Γ the cone generated in Lp/q by Since q ϕ1 , . . . , ϕn , and for Φ the Lp/q -norm):  n λi = 0 ∀λ1 , . . . , λn ∈ IR r

=⇒ (−1)



i=1

 p/q λi1 . . . λi2k ϕi1 + · · · + ϕi2k p/q ≥ 0 .

1≤i1 ,...,i2k ≤n

(b) Sufficiency. Assume that condition (4.2) holds. By Lemma 4.3 we may suppose that X is finite dimensional. As in the proof of Theorem 4.1, we may reduce to the case where X = IRd , equipped with a lattice norm. Let Γ = IRd+ and set 1/q 1/q Φ(x) = x1/q qX (where x1/q = (ξ1 , . . . , ξd ) whenever x = (ξ1 , . . . , ξd )). Then 1/q Φ verifies condition (4.3) with α = p/q (use (4.2) with xi = xi ). By Fact 4.5 there exists a positively linear map T from IRd+ into the positive cone of some space Lp/q (Ω, Σ, µ) such that T xp/q = Φ(x) for every x ∈ IRd+ .

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Let dq be the d-dimensional q -space, and define a linear operator S : IRd+ → Lp (Ω, Σ, µ; dq ) by:  S ξi ei = ξi (T ei )1/q ⊗ fi i

i

where (e1 , . . . , ed ), resp. (f1 , . . . , fd ) is the natural basis of IRd , resp. dq . Note that the images of the basis vectors of IRd are positive and disjoint, hence S is a lattice homomorphism. Moreover: 1/q 1/q d d q q |ξi | T ei = T |ξi | ei = (T |x|q )1/q . Nq (Sx) = i=1

i=1

Hence: 1/q

Sx = Nq (Sx)p = T |x|q p/q = Φ(|x|q )1/q = xX


and thus S is an isometry. Let us now tackle the analogue of Theorem 4.1 with q = ∞.

Theorem 4.6. Let 1 ≤ p < ∞. A Banach lattice X is (isometrically lattice isomorphic to) a sublattice of a space Lp {L∞ } iff ! n λi = 0 ∀n ∈ IN ∀x1 , . . . , xn ∈ X ∀λ1 , . . . , λn ∈ IR

=⇒



"

i=1

λi λj  |xi | ∨ |xj | pX ≤ 0 .

(4.4)

1≤i,j≤n

Condition (4.4) states that the function f → f pX is negative definite on the semigroup (X+ , ∨), in the sense of [BCR], where X+ is the positive cone of X and ∨ the lattice join on X. We will use the following analogues of Lemmas 4.2, 4.3: Lemma 4.7. Every negative definite function f on the semi-group (IRd+ , ∨) which is homogeneous of degree p > 0 (relative to the ordinary scalar multiplication) has the form  p    ui xi  dσ(u) f (x) = B

1≤i≤d

where σ is a Radon measure on some compact base B of the cone IRd+ . Proof. See the Appendix. Lemma 4.8. Let X be a Banach lattice. Assume that each finite dimensional sublattice of X is isometrically lattice isomorphic to a sublattice of an Lp {L∞ }-Banach lattice (allowing the measure spaces to vary). Then there is an isometric lattice isomorphism of X itself into some Lp {L∞ }-Banach lattice.

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Proof. Since the bidual space X ∗∗ is lattice finitely representable in X (see e.g., [B], Theorem 2, or [K¨ u], Theorem 2), every finite-dimensional sublattice Λ of X ∗∗ is, for every ε > 0, (1 + ε)-isomorphically lattice embeddable into some space d Lp ((ΩΛ , ΣΛ , µΛ ); ∞Λ,ε ). Since the Banach lattice X ∗∗ is order-complete, it results from Freudenthal’s Spectral Theorem (and its proof as given in [MN], Theorem ∗∗ the linear subspace Ve of X ∗∗ spanned by the com1.2.18) that for every e ∈ X+ ponents of e is dense in the order ideal generated by e. Note that Ve is a (non closed) vector sublattice of X ∗∗ . From this it is immediate that for every finite dimensional subspace E of X ∗∗ and every positive real number ε there exist a finite-dimensional sublattice ΛE,ε of X ∗∗ and a linear map TE,ε : E → ΛE,ε such that TE,ε x − x ≤ εx for every x ∈ E (indeed, approximate sufficiently  closely each element xi of a basis of E by an element xi of Ve , where e = i |xi |, and note that the linear span of the xi ’s is a sublattice of X ∗∗ ; then set TE,ε xi = xi ). Then ∀x1 , x2 , x3 ∈ E (TE,ε x1 ∨ TE,ε x2 − TE,ε x3 ) − (x1 ∨ x2 − x3 ) ≤ 3ε max xi . i=1,...,n

In particular,

∀x1 , x2 , x3 ∈ E TE,ε x1 ∨ TE,ε x2 − TE,ε x3  − x1 ∨ x2 − x3  ≤ 3ε max xi . i=1,...,n

Let SE,ε be a (1 + )-isomorphic lattice homomorphism from ΛE,ε into some Lp (d∞ )-Banach lattice LE,ε . Let U be an ultrafilter on the set Gf (X ∗∗ ) × (0, 1] contain(where Gf (X ∗∗ ) denotes the sets of finite dimensional subspaces of X ∗∗ )  ing all the sets IE,ε = {(F, η) | F ⊃ E, η ≤ ε}. Let L be the ultraproduct U LE,ε and let U be the diagonal map X ∗∗ → L defined by U x = [SE,ε TE,ε x]U where it is understood that TE,ε x = 0 when x ∈ X ∗∗ \E. We have then ∀x1 , x2 , x3 ∈ X ∗∗ (U x1 ∨ U x2 ) − U x3 ) = (x1 ∨ x2 ) − x3 . With x1 = x2 = 0 we see that U is an isometry; taking x3 = x1 ∨ x2 we have U (x1 ∨ x2 ) = U x1 ∨ U x2 , showing that U is a lattice homomorphism. It is known from [HLR] (Theorem 8.11) that L is a sublattice of some Lp {L∞ }-space. On the other hand X is a sublattice of X ∗∗ (see [MN], Proposition 1.4.5). Therefore X embeds by an isometric lattice isomorphism into some
Lp {L∞ }-space. Proof of Theorem 4.6. (a) Necessity: Assume first that X = IR. For every t, s ∈ IR+ we have:  ∞ 1[0,t] (u)1[0,s] (u)du. t ∨ s = (t + s) − t ∧ s = t + s − 0

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Hence if t1 , . . . , tn ≥ 0: λi λj (ti ∨ tj ) = 1≤i,j≤n





λi λj (ti + tj ) −

1≤i,j≤n



= − ≤ 0.

0

∞

 n

0 2

λi 1[0,ti ] (u)



∞

Positivity

λi λj 1[0,ti ] (u)1[0,tj ] (u)du

1≤i,j≤n

du

i=1

Assume now that X = Lp {L∞ }. Then  p λi λj  |xi | ∨ |xj | X = λi λj N∞ (|xi | ∨ |xi |)p dµ1 Ω1 1≤i,j≤n

1≤i,j≤n

 = ≤ 0



Ω1 1≤i,j≤n

λi λj N∞ (xi )p ∨ N∞ (xj )p dµ1

by the scalar case. (b) Sufficiency: The finite dimensional case follows from Lemma 4.7 and the general case from Lemma 4.8.
Remark 4.9. There is no analogous theorem for p = ∞, q < ∞. In fact it is well known (see e.g. [Lo], Lemma 3.4) that every Banach lattice embeds (isometrically as sublattice) into some ∞ (Γ; L1 (Ω, A, µ)). Using Krivine’s abstract convexification-concavification procedure for Banach lattices (see e.g. [LT], §1.d), which is preserved under lattice isomorphisms, it is then easy to deduce that the class of Banach lattices which are embeddable into some L∞ (Lq )-Banach lattice coincides with that of Banach lattices X which are isometrically q-convex, i.e. such that  ∀x, y ∈ X, (|x|q + |y|q )1/q  ≤ (xq + yq )1/q and that such Banach lattices embed indeed into some ∞ (Γ; Lq (Ω, A, µ))-Banach lattice.

5. Appendix In this appendix we give a proof of Lemmas 4.2 and 4.7. Let (S, +) be an abelian semi-group with neutral element 0. Recall that a semi-character of S is an homomorphism from the semigroup S into the semigroup (IR, ×) preserving the neutral elements. Let 1 be the constant semicharacter. We denote by S the set of bounded semicharacters, which is a compact topological semigroup for operation of pointwise multiplication and the topology of pointwise convergence, with neutral element 1. By [BCR], Theorem 4.3.20 every negative definite function ψ on S which is bounded from below has the form  (1 − ρ(s)) dµ(ρ) ψ(s) = ψ(0) + q(s) +  S\{1}

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where q is an homomorphism from S into the semi-group (IR+ , +) (with q(0) = 0) and µ is a positive Radon measure on the locally compact space S \ {1} which integrates all functions ρ → ρ(s), s ∈ S. This measure is uniquely determined by the function ψ. Hence the homomorphism q is also uniquely determined by ψ, since q(s) = lim ψ(ns)/n. (This result is known as the ‘Levy-Khinchine integral n→∞

representation’ of ψ and µ is the ‘Levy measure’ of ψ.) Proof of Lemma 4.2. Let us first remark that it is sufficient to prove the Lemma when q = 1. In fact, the semi-groups (IRd+ , +q ) and (IRd+ , +) are isomorphic, by the isomorphism θq : x → xq . Every negative definite function f on (IRd+ , +q ) thus has the form f = g ◦ θq , where g is a negative definite function on (IRd+ , +). Moreover, if f is p-homogeneous, then g is p/q-homogeneous. It is then easy to see how to deduce the integral representation of f from that of g. The bounded semicharacters of the semigroup (IRd+ , +) are the generalized d negative exponentials ρa : x → e− a,x where a ∈ [0, +∞]d and a, x = i=1 ai xi , with the convention that (+∞) × t = +∞ if t > 0, = 0 if t = 0, and e−∞ = 0 (see [BCR], 4.4.6). Hence the Levy-Khinchine representation of a negative definite function on (IRd+ , +) is  f (x) = f (0) + q(x) + (1 − e− a,x ) dµ(a). [0,+∞]d\{0}

The integrability requirement on the Levy measure µ is easily seen to be equivalent  d to the condition that 1 ∧ ( i=1 ai ) dµ(a) < ∞. The condition that f is homogeneous of degree p with 0 < p < 1 implies that f (0) = 0 and q(x) = lim f (nx)/n = 0. Moreover, denoting by Hi the set n→+∞

{a ∈ [0, +∞]d \ {0, . . . , 0} | ai = +∞}, we have by the monotone convergence theorem:  d # 0 = lim f (λx) = µ Hi λ→0+

i=1

so we may consider µ as a Radon measure on IRd+ \{0}. For every λ > 0 we have   − a,λx f (λx) = (1 − e ) dµ(a) = (1 − e− λa,x ) dµ(a) IRd \{0} IRd \{0} + +  = (1 − e− a,x ) dµλ (a) IRd +\{0}

where µλ is the image of the measure µ by the scaling x → λx. Since f (λx) = λp f (x) we obtain by the uniqueness of the Levy measure that µλ = λp µ (equivalently, µ(λ−1 A) = λp µ(A) for every Borel subset A of IRd+ \{0}).  Let B be, for example, the simplex IRd+ ∩{a | |a|1 = 1}, where |a|1 = i ai . It is not hard to see that there is a positive Radon measure σ on B such that, in the natural identification of IRd+\{0} with B×(0, +∞) (by the map a → ( |a|a 1 , |a|1 )), the

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dt measure µ gets identified with dσ ⊗ tp+1 . (Take, in fact, σ(A) = p · µ(A × [1, +∞)) for every Borel subset A of B). Then:   +∞  dt f (x) = (1 − e− tu,x ) p+1 dσ(u) = Cp u, xp dσ(u) t B 0 B  ∞ du where Cp = (1 − e−u ) p+1
u 0

Proof of Lemma 4.7. The semi-characters of the idempotent semi-group (IRd , ∨) are exactly the characteristic functions of the left-hereditary sub-semigroups (see [BCR], Proposition 4.4.17). In the case d = 1, such left-hereditary subsemigroups S are exactly the intervals [0, a] or [0, a) (or [0, +∞)). In the case d > 1, these left-hereditary subsemigroups are the products of d such 1-dimensional intervals. We have Sd = (S1 )d (with the product topology) and S1 identifies with the twopoint compactification S1 of the product space (0, +∞) × {0, 1} equipped with the lexicographic order topology. This identification is the following: 1{0} → 0 1[0,a) → a− = (a, 0) 1[0,a] → a+ = (a, 1) 1[0,+∞) → +∞

 for all 0 < a < +∞

We put 0± = 0 and (+∞)± = +∞. The Levy-Khinchine integral representation of a negative definite function f % $ on IRd+ , ∨ (which is automatically bounded from below) is then: f (x) = f (0) + µ({t ∈ S1d | ∃i = 1, . . . , d : 0 ≤ ti < x+ i }). Note that there is no homomorphism q : (IRd+ , ∨) → (IR+ , +) except the trivial one (0). Let x ∈ IRd , x = 0. When λ → 1+ we have by the monotone convergence theorem: f (λx) = f (0) + µ({t ∈ S1d | ∃i = 1, . . . , d : 0 ≤ ti < (λxi )+ }) → f (0) + µ({t ∈ S1d | ∃i = 1, . . . , d : 0 ≤ ti ≤ x+ i = 0}). When λ → 0 we obtain: f (λx) → f (0) + µ({t ∈ S1d | ∃i = 1, . . . , d : ti = 0 = x+ i }). Similarly when λ → 1− we find that: f (λx) → f (0) + µ({t ∈ S1d | ∃i = 1, . . . , d : 0 ≤ ti < x− i }). Hence f is radially continuous iff µ({t ∈ S1d | ti ∈ {a− , a+ }}) = 0 for every a ∈ [0, +∞) and every i = 1, . . . , d. (Recall that f is said to be radially continuous iff f (λx) → f (λ0 x) for every x ∈ IRd+ when λ → λ0 in IR+ .) Let q be the continuous surjective map S1d → [0, +∞]d defined by q(a) = ± (a1 , . . . , ad ) when a = (a± 1 , . . . , ad ), and let ν = qµ be the image of the measure µ

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by this map (ν is a Radon measure on the open set [0, +∞]d \{(+∞, . . . , +∞)}). We then have: f (x) = f (0) + ν({t ∈ [0, +∞]d | ∃i = 1, . . . , d, 0 ≤ ti < xi }) and ν({t ∈ [0, +∞]d | ti = a}) = 0 for every 0 ≤ a < +∞ and i = 1, . . . , d. It is clear that the measure ν is determined by f . Consider the homeomorphism J : [0, +∞]d → [0, +∞]d , (t1 , . . . , td ) → −1 −1 = +∞, +∞−1 = 0. The image ρ = Jν (t1 , . . . , t−1 d ), with the convention 0 of the measure ν is a Radon measure on [0, +∞]d \ {0, . . . , 0}, such that ρ({t ∈ [0, +∞]d | ti = +∞}) = 0 for every i = 1, . . . , d. Hence it is a Radon measure on [0, +∞)d \ {0, . . . , 0} which integrates all sets that are complements of open neighborhoods of the origin. We then have: f (x) = f (0) + ρ({t ∈ IRd+ \{(0, . . . , 0)} | ∃i = 1, . . . , d, ti ≥ x−1 i })  = f (0) + ρ({t ∈ IRd+ \{(0, . . . , 0)} | ti xi ≥ 1}). 1≤i≤d

Assume now that f is homogeneous of degree p; then so is ρ, in the sense that ρ(λ−1 A) = λp ν(A) for every Borel set A; again ρ identifies with the measure dr σ⊗ p+1 under the homeomorphism IRd+\{(0, . . . , 0)} → (0, +∞)×, x → ( |x|x 1 , |x|1 ). r Hence we have:  p     dr 1  f (x) = dσ(u) = ui xi  dσ(u).  p+1 p B B r( ui xi )≥1 r 1≤i≤d


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C. Ward Henson Math. Dept., University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana Illinois 61801 USA Yves Raynaud Institut de Math´ematiques de Jussieu (CNRS) Projet Analyse Fonctionnelle Case 186 4, place Jussieu 75252 Paris Cedex 05 France Received 3 February 2006; accepted 21 July 2006