MARTINGALES IN BANACH LATTICES

VLADIMIR G. TROITSKY

Dedicated to the memory of Yuri Abramovich, my friend and advisor.

Abstract. In this article we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (E_{n}) on a Banach lattice F is said to be a filtration if E_{n}E_{m} = E_{n∧m}. A sequence (x_{n}) in F is a martingale if E_{n}x_{m} = x_{n} whenever n ⩽ m. Denote by M = M(F, (E_{n})) the Banach space of all norm uniformly bounded martingales. It is shown that if F doesn’t contain a copy of c_{0} or if every E_{n} is of finite rank then M is itself a Banach lattice. Convergence of martingales is investigated and a generalization of Doob Convergence Theorem is established. It is proved that under certain conditions one has isometric embeddings F ↪ M ↪ F^{∗∗}. Finally, it is shown that every martingale difference sequence is a monotone basic sequence.

In this paper we define a martingale in terms of Banach lattices. Rephrasing the title of [Wil91], this paper could be entitled Martingales without probability. We start with a brief review of classical martingale theory and Banach lattices.

1. Classical martingale theory

In the following, we fix a probability space (Ω, F, P) and a filtration (F_{n})_{n=1}^{∞}, i.e., an increasing sequence of sub-sigma-algebras of F. It is often convenient to assume that F = ⋁_{n=1}^{∞} F_{n}, as otherwise one can replace F with ⋁_{n=1}^{∞} F_{n}. We will write L_{p}(P) for L_{p}(Ω, F, P).

A sequence X = (x_{n})_{n=1}^{∞} of functions in L_{1}(P) is called a martingale relative to (F_{n}) and P if E(x_{m} | F_{n}) = x_{n} whenever m ⩾ n, and a submartingale if E(x_{m} | F_{n}) ⩾ x_{n} whenever m ⩾ n. A martingale X is L_{p}-bounded if its L_{p}-martingale norm, given by ∥X∥_{p} = sup_{n}∥x_{n}∥_{p}, is finite. Let the symbol M_{p} = M_{p}(Ω, F, (F_{n}), P ) denote the space of all L_{p}-bounded martingales.

Date: March 30, 2004. 2000 Mathematics Subject Classification. 60G48, 46B42. Key words and phrases. Martingale, filtration, Banach lattice.

Positivity, to appear.

1 2 V. G. TROITSKY

A martingale X is called uniformly integrable if for every positive ε there exists a number K such that

∫_{|x_{n}|>K} |x_{n}| dP < ε

for every n. Doob Convergence Theorem [Doob53] asserts that a martingale X is uniformly integrable if and only if converges in L_{1}-norm to some function x ∈ L_{1}(P). In this case x_{n} = E(x | F_{n}) and ∥x∥_{1} = lim_{n}∥x_{n}∥_{1} = ∥X∥. Thus, the set of all uniformly integrable martingales is a proper subspace of M_{1}, isometrically isomorphic to L_{1}(P), the isomorphism maps X to x. For example, the so called double-or-nothing martingale x_{n} = 2^{n}χ_{[0,2^{−n})} defined on the unit interval endowed with Lebesgue measure and dyadic filtration is L_{1}-bounded but not uniformly integrable. However, for p > 1 every L_{p}-bounded martingale is uniformly integrable, so that M_{p} is isometrically isomorphic to L_{p}(P). For further details on classical theory of martingales see [Wil91, Doob53, Doob94].

2. Banach lattices

A vector lattice is a vector space equipped with a lattice order relation, which is compatible with the linear structure. A Banach lattice is a vector lattice with a Banach norm which is monotone, i.e., 0 ⩽ x ⩽ y implies ∥x∥ ⩽ ∥y∥, and satisfies ∥|x|∥ = ∥x∥ for every two vectors x and y, where |x| = x ∨ (−x). The spaces C(K) and L_{p}(μ) for 1 ⩽ p ⩽ +∞ are important examples of Banach lattices. A Banach lattice where ∥x + y∥ = ∥x∥ + ∥y∥ holds for every two nonnegative vectors x and y is called an abstract L_{1}-space or AL-space. A vector lattice is said to be Dedekind complete if every nonempty subset that is bounded above has a supremum. We say that a Banach lattice has order continuous norm if ∥x_{α}∥ → 0 for every decreasing net (x_{α}) with inf_{α} x_{α} = 0. A Banach lattice with order continuous norm is Dedekind complete. A Banach lattice is called a Kantorovič-Banach space or a KB-space whenever every increasing norm bounded sequence (x_{n}) is norm convergent. In this case norm continuity of lattice operations implies that lim_{n} x_{n} = sup_{n} x_{n}. It is known that a Banach lattice F is a KB-space iff c_{0} is not embeddable in F iff F is weakly sequentially complete. In particular, reflexive Banach lattices and AL-spaces are KB-spaces. The norm in a KB-space is order continuous. A sublattice E of a vector lattice is called an (order) ideal if y ∈ E and |x| ⩽ |y| imply x ∈ E. An ideal E is called a band if x = sup_{α} x_{α} implies x ∈ E for every positive increasing net (x_{α}) in E. Two elements x and y in a vector lattice are said to be disjoint (in symbols x ⊥ y) whenever |x| ∧ |y| = 0 holds. If A is a nonempty subset of a vector lattice, then its disjoint complement A^{d} is the

MARTINGALES IN BANACH LATTICES 3

set of all elements of the lattice, disjoint to every element of A. A band E in a vector lattice F that satisfies F = E ⊕ E^{d} is reffered to as a projection band. Every band in a Dedekind complete vector lattice is a projection band. An operator T on a Banach lattice F is positive if it preserves the cone F_{+} of positive elements. It is easy to see that T ⩾ 0 iff x ⩽ y implies T x ⩽ T y; furthermore, if T ⩾ 0 then T (x∨y) ⩾ (T x)∨(T y) and |T x| ⩽ T |x| for any x, y ∈ F. See [AB85, AA02, LZ71, Zaan83, MN91] for more details on Banach lattices. Throughout the rest of the paper F will be a fixed Banach lattice.

3. Main definitions

We will introduce a generalization of the concept of a martingale. By a martingale we will mean a sequence of elements in a Banach lattice satisfying certain properties. Douglas proved in [Doug65] (see also [AAB93]) that the conditional expectations are the only contractive projections on L_{1}(P) preserving constant functions. In view of this result of Douglas it will be natural in our setting to replace conditional expectations with positive contractive projections. This argument justifies the following notation. A sequence of positive contractive projections (E_{n}) on a Banach lattice F is called a filtration if E_{n}E_{m} = E_{n∧m}. Note that the ranges of E_{n}’s form a nested increasing sequence. Here E_{n}’s play the role of conditional expectations.

A filtration is said to be dense if E_{n}x → x in norm for every x ∈ F, or, equivalently, if ⋃_{n=1}^{∞} Range E_{n} is dense in F. This is analogous to the condition F = ⋁_{n=1}^{∞} F_{n} for classical filtrations. But, unlike in the case of classical filtration, generally we cannot simply replace F with \overline{⋃_{n=1}^{∞} Range E_{n}} because the latter set need not be a Banach sublattice of F.

Note, however, that if a projection P is strictly positive, i.e., P x > 0 whenever x > 0, then Range P is a sublattice of F (see [Sch74, dJ82]). Therefore, if E_{m} is strictly positive for some m then \overline{⋃_{n=1}^{∞} Range E_{n}} is a Banach sublattice of F. Indeed, for every x > 0 and every n ⩾ m we have E_{n}x ≠ 0 because 0 < E_{m}x = E_{m}E_{n}x. It follows that E_{n} is strictly positive for all n ⩾ m, so that Range E_{n} is a sublattice. Thus, when a filtration contains a strictly positive projection, one can assume that it is dense by replacing F with \overline{⋃_{n=1}^{∞} Range E_{n}}.

A sequence X = (x_{n}) of elements of F is called a martingale relative to a filtration (E_{n}) if E_{n}x_{m} = x_{n} whenever n ⩽ m. It follows, in particular, that x_{n} ∈ Range E_{n} for every n. A sequence X = (x_{n}) is called a submartingale if E_{n}x_{m} ⩾ x_{n} whenever n ⩽ m. A (sub)martingale is said to be bounded if its martingale norm given by

4 V. G. TROITSKY

∥X∥ = sup_{n}∥x_{n}∥ is finite. Notice that if (x_{n}) is a martingale or a positive submartingale, then the sequence ∥x_{n}∥ is increasing. Indeed, ∥x_{n}∥ ⩽ ∥E_{n}x_{n+1}∥ ⩽ ∥x_{n+1}∥. Therefore, ∥X∥ = lim_{n}∥x_{n}∥.

Denote by M = M(F, (E_{n})) the class of all bounded martingales. Note that M is a closed subspace of the space (⨁_{n=1}^{∞} F )_{∞}, hence M is a Banach space.

4. Examples

In the case of classical martingales we have F = L_{1}(P) and E_{n} = E(· | F_{n}). It was mentioned in the very beginning that one can usually assume that the filtration is dense. Observe, that a classical filtration satisfies another important property: it preserves norms of positive vectors, that is, ∥E_{n}x∥ = ∥x∥ for every n and every x ∈ F_{+}.

Next, we present several examples of filtrations and martingales related to bases in Banach spaces. For the terminology related to bases in Banach spaces we refer the reader to [LT77].

Example 1. Suppose that (e_{i}) is a 1-unconditional basis in F such that ∑_{n=1}^{∞} α_{i}e_{i} ⩾ 0 iff α_{i} ⩾ 0 for all i ⩾ 1. For every n ⩾ 1 let E_{n} be the n-th basis projection given by E_{n}(∑_{i=1}^{∞} α_{i}e_{i}) = ∑_{i=1}^{n} α_{i}e_{i}, then (E_{n}) is a dense filtration. Notice that (x_{n}) is a martingale iff there exists a sequence of scalars (α_{i}) such that x_{n} = ∑_{i=1}^{n} α_{i}e_{i} for every n ⩾ 1. A martingale (x_{n}) is convergent if and only there exists z = ∑_{i=1}^{∞} α_{i}e_{i} such that x_{n} = ∑_{i=1}^{n} α_{i}e_{i}, in this case x_{n} converges to z. The basis is boundedly complete if and only if every bounded martingale converges.

Example 2. Consider the special case of Example 1 when F = c_{0}. Again, X = (x_{n}) is a martingale in c_{0} iff there exists a sequence of scalars (α_{i}) such that x_{n} = ∑_{i=1}^{n} α_{i}e_{i} for every n ⩾ 1. Notice that X is bounded if and only if the sequence (α_{i}) is bounded, and in this case ∥X∥ = sup_{i}|α_{i}|. Thus, in this case M can be identified with ℓ_{∞}. Observe that here M is non-separable even though X is separable.

Example 3. Let F = C[0, 1] and (e_{i}) be the Schauder system in C[0, 1]. Again, let E_{n} be the n-th basis projection. One can easily see that for f ∈ C[0, 1] its image E_{n}f agrees with f on a set of dyadic points and is linear between those points. In particular, every E_{n} is a positive operator. Since the Schauder system is a monotone basis, each E_{n} is a contraction. Hence, (E_{n}) is a dense filtration on F. Clearly, not every martingale is convergent. For example, put x_{n}(0) equal 0 at 0, equal 1 at all the other dyadic points corresponding to E_{n}, and linear in between. Then (x_{n}) is a non-convergent martingale.

MARTINGALES IN BANACH LATTICES 5

Example 4. Let F = L_{p}[0, +∞) (1 ⩽ p < +∞) and put E_{n}x = x · χ_{[0,n]}, i.e., E_{n} “cutsoff” the tail of x after n. One can easily see that (E_{n}) is a dense filtration. A sequence (x_{n}) is a martingale if x_{n} = ∑_{i=1}^{n} h_{i} where (h_{i}) is a sequence in F such that supp h_{i} ⊆ [i−1, i]. It can be easily verified that the map x ∈ F → (x · χ_{[0,n]}) is an isometry from F onto M.

5. When is M a Banach lattice?

Introduce an order on (⨁_{n=1}^{∞} F )_{∞} as follows: if X = (x_{n}) and Y = (y_{n}), we say that X ⩽ Y if x_{n} ⩽ y_{n} for each n. With this order M is an ordered Banach space. Clearly, the norm is monotone: if 0 ⩽ X ⩽ Y then ∥X∥ ⩽ ∥Y ∥.

Notice that if the filtration preserves the norms of positive vectors and X = (x_{n}) is a positive martingale then ∥x_{n}∥ = ∥E_{n}x_{m}∥ = ∥x_{m}∥ whenever n ⩽ m, so that ∥x_{n}∥ = ∥X∥ for every n. Thus, in this case every positive martingale is bounded.

It is not immediately obvious if M is a lattice in the order we just introduced, and how one could compute X ∨ Y, X ∧ Y, and |X| for two martingales X = (x_{n}) and Y = (y_{n}) in M. The “natual guess” that X ∨ Y = (x_{n} ∨ y_{n})_{n=1}^{∞}, X ∧ Y = (x_{n} ∧ y_{n})_{n=1}^{∞}, and |X| = (|x_{n}|)_{n=1}^{∞} turns out to be wrong. Even when (x_{n}) is a martingale, (|x_{m}|) doesn’t have to be a martingale. For example, let (Ω, F, P) be the unit segment endowed with Lebesgue measure and (F_{n}) be the dyadic filtration. Consider an L_{1}-bounded martingale defined in the following way: x_{1} ≡ 0, x_{n} = χ_{[0,1/2)} − χ_{[1/2,1]} for n > 1. It is easy to see that (|x_{n}|) is not a martingale. Notice, however, that if (x_{n}) and (y_{n}) are two submartingales then (x_{n} ∨ y_{n}) is a submartingale. Indeed, if n ⩾ m then E_{m}(x_{n} ∨ y_{n}) ⩾ (E_{m}x_{n}) ∨ (E_{m}y_{n}) ⩾ (x_{m} ∨ y_{m}). In particular, if (x_{n}) is a martingale, then (|x_{n}|) is a submartingale. Lemma 5. Let X = (x_{n}) and Y = (y_{n}) be two bounded submartingales.

(i) For a fixed n, the sequence (E_{n}(x_{m} ∨ y_{m}))_{m=n}^{∞} is increasing, norm bounded by ∥X∥ + ∥Y ∥, and bounded below by x_{n} ∨ y_{n}.

(ii) If, in addition, this sequence converges in norm to some (z_{n}) for each n, then Z = (z_{n}) is a martingale, and it is the least martingale satisfying X ⩽ Z and Y ⩽ Z.

Proof. Let X = (x_{n}) and Y = (y_{n}) be two bounded submartingales and n ⩽ m, notice that E_{n}(x_{m} ∨ y_{m}) ⩾ (E_{n}x_{m}) ∨ (E_{n}y_{m}) ⩾ x_{n} ∨ y_{n}. Furthermore,

E_{n}(x_{m+1} ∨ y_{m+1}) = E_{n}E_{m}(x_{m+1} ∨ y_{m+1}) ⩾ E_{n}(E_{m}x_{m+1} ∨ E_{m}y_{m+1}) ⩾ E_{n}(x_{m} ∨ y_{m}).

Finally,

∥E_{n}(x_{m} ∨ y_{m})∥ ⩽ ∥x_{m} ∨ y_{m}∥ ⩽ ∥|x_{m}| + |y_{m}|∥ ⩽ ∥X∥ + ∥Y ∥.

6 V. G. TROITSKY

Suppose that lim_{m} E_{n}(x_{m} ∨ y_{m}) = z_{n} for each n, and set Z = (z_{n}). First, observe that Z is a martingale. Indeed, for k ⩽ n we have

E_{k}z_{n} = E_{k}( lim_{m→∞} E_{n}(x_{m} ∨ y_{m})) = lim_{m→∞} E_{k}E_{n}(x_{m} ∨ y_{m}) = lim_{m→∞} E_{k}(x_{m} ∨ y_{m}) = z_{k}.

Since E_{n}(x_{m} ∨ y_{m}) ⩾ x_{n} ∨ y_{n} whenever m ⩾ n, we have z_{n} ⩾ x_{n} ∨ y_{n} for all n. Thus, Z ⩾ X and Z ⩾ Y. On the other hand, suppose that Z̃ = (z̃_{n}) is a martingale such that Z̃ ⩾ X and Z̃ ⩾ Y. Then z̃_{m} ⩾ x_{m} ∨ y_{m} for all m, so that z̃_{n} = E_{n}z̃_{m} ⩾ E_{n}(x_{m} ∨ y_{m}) for all m ⩾ n. As lim_{m} E_{n}(x_{m} ∨ y_{m}) = z_{n}, this yields z̃_{n} ⩾ z_{n}, so that Z̃ ⩾ Z. □

Lemma 6. Let X ∈ M such that the limit lim_{m} E_{n}|x_{m}| exists for each n, denote it z_{n}. Put Z = (z_{n}), then Z is a martingale, Z = |X|, and ∥Z∥ = ∥X∥.

Proof. Apply Lemma 5 to X and −X, it follows that Z is indeed a bounded martingale and Z = |X|. Notice that

∥z_{n}∥ = lim_{m→∞}∥E_{n}|x_{m}|∥ ⩽ lim_{m→∞}∥x_{m}∥ = ∥X∥.

On the other hand, for n ⩽ m we have |x_{n}| = |E_{n}x_{m}| ⩽ E_{n}|x_{m}|. It follows that

∥z_{n}∥ = lim_{m→∞}∥E_{n}|x_{m}|∥ ⩾ ∥x_{n}∥.

This yields ∥Z∥ = ∥X∥. □

Theorem 7. If F is a KB-space then M = M(F, (E_{n})) is a Banach lattice with lattice operations given by

(X ∨ Y)_{n} = lim_{m→∞} E_{n}(x_{m} ∨ y_{m})
(X ∧ Y)_{n} = lim_{m→∞} E_{n}(x_{m} ∧ y_{m})
(X^{+})_{n} = lim_{m→∞} E_{n}(x_{m}^{+})
(X^{−})_{n} = lim_{m→∞} E_{n}(x_{m}^{−})
|X|_{n} = lim_{m→∞} E_{n}|x_{m}|

(1)

Proof. Let X = (x_{n}) and Y = (y_{n}) be two bounded martingales in M. It follows from Lemma 5(i) that for every n the sequence (E_{n}(x_{m} ∨y_{m}))_{m=n}^{∞} is increasing in m and norm bounded, hence it converges. It follows then by Lemma 5(ii) that X ∨ Y exists and is given by (X ∨ Y)_{n} = lim_{m→∞} E_{n}(x_{m} ∨ y_{m}). The other formulae in (1) follow immediately. This proves that M is a lattice. Finally, ∥|X|∥ = ∥X∥ by Lemma 6, so that M is a Banach lattice. □

Corollary 8. If F is an AL-space and (E_{n}) is a filtration on F, then M = M(F, (E_{n})) is an AL-space.

MARTINGALES IN BANACH LATTICES 7

Proof. Theorem 7 yeilds that M is a Banach lattice. Suppose that X, Y ∈ M_{+}, X = (x_{n}), and Y = (y_{n}). Then x_{n}, y_{n} ∈ F_{+} for each n, so that ∥x_{n} + y_{n}∥ = ∥x_{n}∥ + ∥y_{n}∥. It follows that

∥X + Y ∥ = lim_{n→∞}∥x_{n} + y_{n}∥ = lim_{n→∞}(∥x_{n}∥ + ∥y_{n}∥) = ∥X∥ + ∥Y ∥.

□

In the case of classical martingales, it was shown by Krickeberg [Kr56] that the space M_{1} of all L_{1}-bounded martingales in L_{1}(P) is a Dedekind complete vector lattice. In particular, every L_{1}-bounded martingales can be written as a difference of two positive martingales. Corollary 8 yields the following refinement of Krickeberg’s result. Corollary 9. The space M_{1} of all L_{1}-bounded martingales in L_{1}(P) is a Banach lattice and, moreover, an AL-space.

Next, we show that if a filtration consists of finite rank operators, then the space of bounded martingales is a Banach lattice. Lemma 10. An increasing norm bounded sequence contained in a finite-dimensional subspace of a Banach lattice has a supremum and converges to it in norm.

Proof. Suppose that (x_{n}) is an increasing sequence contained in the unit ball B_{E} of a finite-dimensional subspace E of a Banach lattice. By the continuity of lattice operations, x_{n} ⩽ lim_{i} x_{n_{i}} for every n ⩾ 0 and every convergent subsequence (x_{n_{i}}). Therefore, if x_{m_{i}} is another convergent subsequence then lim_{i} x_{m_{i}} ⩽ lim_{i} x_{n_{i}}. It follows that all convergent subsequences of (x_{n}) have the same limit. Compactness of B_{E} completes the proof. □

Proposition 11. If F is a Banach lattice and (E_{n}) is a filtration on F such that E_{n} is of finite rank for each n, then M = M(F, (E_{n})) is a Banach lattice with lattice operations given by (1).

Proof. Let X = (x_{n}) and Y = (y_{n}) be two martingales in M. It follows from Lemma 5(i) that for every n the sequence (E_{n}(x_{m} ∨ y_{m}))_{m=n}^{∞} is increasing in m. This sequence is contained in the range of E_{n}, so that it converges by Lemma 5(ii), and if we denote the limit by z_{n} then Z = (z_{n}) is a martingale and Z = X ∨ Y. This proves the first formula in (1). The other formulae follow easily, so that M is a lattice. Finally, Lemma 6 yields that M is a Banach lattice. □

Observe that the filtrations in Examples 1–3 consist of finite-rank projections. It follows that bounded martingales in those examples form Banach lattices.

8 V. G. TROITSKY

6. Regular martingales

Given a martingale X in a Banach lattice F, we say that X is regular if there exists a positive martingale Y such that X ⩽ Y. It is easy to see that a martingale is regular if and only if it is a difference of two positive martingales. We denote by M_{r} = M_{r}(F, (E_{n})) the set of all regular bounded martingales. Clearly, M_{r} is a linear subspace of M. It follows from Theorem 7 that if F is a KB-space then for every bounded martingale X in F we have X ⩽ |X|, so that X is regular, hence M_{r} = M. Example 12. A non-regular martingale. Let F = L_{∞}[0, 1]. For every n define E_{n} as follows:

E_{n}x(t) = \begin{cases}the average of x on [0, 2^{−n}] when t ⩽ 2^{−n}; \\ x(t) otherwise. \end{cases}

It is easy to see that (E_{n}) is a filtration. Put D_{n} = [1/2^{n}, 1/2^{n−1}], let A_{n} and B_{n} be the left and the right halves of D_{n} respectively. Set x_{m} = ∑_{k=1}^{m} 2^{k}(χ_{A_{k}} − χ_{B_{k}}), then X = (x_{m}) is an unbounded martingale. Show that there is no positive martingale that dominates X. Indeed, suppose that Y = (y_{n}) is a positive martingale such that X ⩽ Y. Then y_{n} ⩾ x_{n}^{+} for each n. It follows that y_{1} = E_{1}y_{n} ⩾ E_{1}x_{n}^{+}. Notice that x_{n}^{+} = ∑_{k=1}^{n} 2^{k}χ_{A_{k}}, so that the restriction of x_{n}^{+} to [0, 1/2] is ∑_{k=2}^{n} 2^{k}χ_{A_{k}}, which has the average value of n − 1 because the length of A_{k} is 1/2^{k+1}. It follows that E_{1}x_{n}^{+} equals n − 1 on [0, 1/2], so that ∥y_{1}∥ ⩾ ∥E_{1}x_{n}^{+}∥ ⩾ n − 1. It follows that y_{1} ∉ L_{∞}[0, 1], contradiction. Theorem 13. If F has order continuous norm then M_{r} = M_{r}(F, (E_{n})) is a Banach lattice with lattice operations given by (1).

Proof. Let X = (x_{n}) and Y = (y_{n}) be two martingales in M_{r}. Then there exist two positive martingales X̃ = (x̃_{n}) and Ỹ = (ỹ_{n}) such that X ⩽ X̃ and Y ⩽ Ỹ. It follows from Lemma 5(i) that for every n the sequence (E_{n}(x_{m} ∨ y_{m}))_{m=n}^{∞} is increasing in m. Moreover,

E_{n}(x_{m} ∨ y_{m}) ⩽ E_{n}(x̃_{m} ∨ ỹ_{m}) ⩽ E_{n}(x̃_{m} + ỹ_{m}) = x̃_{n} + ỹ_{n},

so that the sequence is order bounded. Since F has order continuous norm (and, in particular, is Dedekind complete), the sequence has a supremum, to which it converges in norm. Call the supremum z_{n}, let Z = (z_{n}). By Lemma 5(i) ∥z_{n}∥ ⩽ ∥X∥ + ∥Y ∥. It follows from Lemma 5(ii) that Z is a bounded martingale (and it is regular because Z ⩽ X̃ + Ỹ) and that Z = X ∨ Y. This proves the first formula in (1). The other formulae follow easily. In particular, M_{r} is a lattice. Lemma 6 yields ∥|X|∥ = ∥X∥ for every X ∈ M_{r}, so that the norm on M_{r} is a lattice norm.

MARTINGALES IN BANACH LATTICES 9

It is left to show that M_{r} is a Banach space. It suffices to prove that M_{r} is a closed subspace of M. Suppose that (X^{(n)})_{n=1}^{∞} is a sequence of regular martingales such that X^{(n)} converges in norm to some X ∈ M, show that X is regular. Without loss of generality, ∥X^{(n+1)} −X^{(n)}∥ < 2^{−n} for every n. Since X^{(n+1)} −X^{(n)} is regular, its modulus exists and

∥|X^{(n+1)} − X^{(n)}|∥ = ∥X^{(n+1)} − X^{(n)}∥ < 2^{−n},

so that the series ∑_{n=1}^{∞}|X^{(n+1)} − X^{(n)}| converges in M. Put Y = ∑_{n=1}^{∞}|X^{(n+1)} − X^{(n)}|. Clearly, Y ⩾ 0. Now

X^{(m)} = X^{(1)} + ∑_{n=1}^{m−1}(X^{(n+1)} − X^{(n)}) ⩽ X^{(1)} + ∑_{n=1}^{m−1}|X^{(n+1)} − X^{(n)}| ⩽ |X^{(1)}| + Y.

It follows that X ⩽ |X^{(1)}| + Y, hence X ∈ M_{r}. □

7. Bounded below filtrations

We say that a filtration (E_{n}) is bounded below on F_{+} if there exists n ⩾ 1 and a constant c > 0 such that ∥E_{n}x∥ ⩾ c∥x∥ for every x ⩾ 0. In particular, if (E_{n}) preserves the norms of positive vectors, then it is bounded below on F_{+}. Observe that if (E_{n}) is bounded below then every positive (and, therefore, every regular) martingale is bounded. Indeed, ∥x_{m}∥ ⩽ \frac{1}{c}∥E_{n}x_{m}∥ = \frac{1}{c}∥x_{n}∥ for every m ⩾ n, so that ∥X∥ ⩽ \frac{1}{c}∥x_{n}∥ < +∞. Notice also that if (E_{n}) is bounded below then E_{n} is strictly positive for some n, so that we can assume without loss of generality that the filtration is dense. Proposition 14. If F is a KB-space and (E_{n}) is bounded below on F_{+}, then M = M(F, (E_{n})) is again a KB-space.

Proof. It follows from Theorem 7 that M is a Banach lattice. Consider an increasing sequence of martingales (X^{(k)})_{k=1}^{∞} with uniformly bounded norms ∥X^{(k)}∥ ⩽ K. Let X^{(k)} = (x_{n}^{(k)}). Then for each n the sequence (x_{n}^{(k)})_{k=1}^{∞} is increasing in k and is norm bounded by K. Since F is a KB-space, this sequence converges to some x_{n} and ∥x_{n}∥ ⩽ K. Let X = (x_{n}), then for n ⩽ m we have

E_{n}x_{m} = E_{n}( lim_{k→∞} x_{m}^{(k)}) = lim_{k→∞} E_{n}x_{m}^{(k)} = lim_{k→∞} x_{n}^{(k)} = x_{n},

so that X ∈ M. Clearly, X = sup_{k} X^{(k)}, hence Proposition 15 implies that ∥X−X^{(k)}∥ → 0. □

Proposition 15. If F has order continuous norm and (E_{n}) is bounded below on F_{+} then the martingale norm is again order continuous.

10 V. G. TROITSKY

Proof. Suppose that (X^{(α)}) is a decreasing net of martingales, X^{(α)} = (x_{n}^{(α)}), such that inf_{α} X^{(α)} = 0. For a fixed n, the net (x_{n}^{(α)})_{α} is positive and decreasing in α, so that order continuity of norm in F yields lim_{α} x_{n}^{(α)} exists, call it x_{n}. It is easy to see that x_{n} ⩾ 0 and that X = (x_{n}) is a martingale:

E_{m}x_{n} = E_{m}(lim_{α} x_{n}^{(α)}) = lim_{α} E_{m}x_{n}^{(α)} = lim_{α} x_{m}^{(α)} = x_{m}.

Clearly, X is bounded and it follows from 0 ⩽ X ⩽ X^{(α)} for all α and inf_{α} X_{(α)} = 0 that X = 0. Thus, lim_{α} x_{n}^{(α)} = 0 for each n.

There exists n ⩾ 1 and a constant c > 0 such that ∥E_{n}x∥ ⩾ c∥x∥ for every x ⩾ 0. For every m ⩾ n we have ∥x_{n}^{(α)}∥ = ∥E_{n}x_{m}^{(α)}∥ ⩾ c∥x_{m}^{(α)}∥, so that ∥X^{(α)}∥ ⩽ \frac{1}{c}∥x_{n}^{(α)}∥ → 0. □

8. A submartingale is dominated by a martingale

We claim that every bounded submartingale in a KB-space is dominated by a unique martingale (which justifies the term submartingale). Suppose that X = (x_{n}) is a bounded submartingale on a KB-space F. Applying Lemma 5(i) with Y = X we see that for a fixed n the sequence (E_{n}x_{m})_{m=n}^{∞} is increasing and norm bounded by ∥X∥. Since F is a KB-space, it converges to some z_{n}, let Z = (z_{n}). By Lemma 5(ii) Z is a martingale; it is the least martingale such that X ⩽ Z. Notice that

∥z_{n}∥ = lim_{m→∞}∥E_{n}x_{m}∥ ⩽ lim_{m→∞}∥x_{m}∥ = ∥X∥,

so that Z is bounded and ∥Z∥ ⩽ ∥X∥. In general, ∥Z∥ need not equal ∥X∥. Even in the real-valued case, let X be a submartingale of constant functions x_{n} = −\frac{1}{n}χ_{[0,1]}, then ∥X∥ > 0 but Z = 0. Nevertheless, if (E_{n}) preserves the norms of positive vectors and X is a positive submartingale then ∥z_{n}∥ = lim_{m}∥E_{n}x_{m}∥ = lim_{m}∥x_{m}∥ = ∥X∥ for all n, so that ∥Z∥ = ∥X∥. Finally, notice that the condition ∥X∥ < ∞ can be replaced by sup_{n}∥x_{n}^{+}∥ < ∞.

9. Convergent martingales

Let F be a Banach lattice and (E_{n}) a filtration on F, denote M = M(F, (E_{n})). One can easily verify that for every x ∈ F the sequence X = (E_{n}x)_{n=1}^{∞} is a martingale. Martingales of this form will be called fixed. Note that ∥X∥ ⩽ ∥x∥, so that the map x ↦ (E_{n}x) is a linear contraction from F into M.

Observe that every convergent martingale is fixed (on its limit). Indeed, if lim_{m} x_{m} = x then lim_{m} E_{n}x_{m} = E_{n}x, but E_{n}x_{m} = x_{n} when m ⩾ n, so that E_{n}x = x_{n}.

Suppose that, in addition, the filtration is dense, that is, E_{n}x → x for every x ∈ F. Then a martingale is fixed iff it is convergent. In this case ∥X∥ = lim_{n}∥x_{n}∥ =

MARTINGALES IN BANACH LATTICES 11

lim_{n}∥E_{n}x∥ = ∥x∥, hence the map x ↦ (E_{n}x) is an isometry. Thus, F isometrically embeds in M. With a slight abuse of notation, we will consider F as a subspace of M.

Furthermore, suppose that F is a KB-space. Let (x_{n}) be an increasing sequence in F_{+}, order bounded in M. Then it is norm bounded and, since F is a KB-space, it has a supremum in F. Since M is a KB-space, hence Dedekind complete, it follows that F is a projection band in M: Proposition 16. If F is a KB-space and (E_{n}) is dense then F is a projection band in M(F, (E_{n})).

In the case of classical martingales this yields that uniformly integrable martingales form a projection band in M_{1}. It would be interesting to describe the disjoint complement of L_{1}(P) in M_{1}, and, more generally, the describe the disjoint complement of F in M under the hypotheses of Proposition 16.

On the other hand, Example 2 shows that if F is not KB, then it doesn’t even have to be complemented in M.

10. Uniform integrability

Dunford-Pettis Theorem [DP40] asserts that a subset of L_{1}(P) is uniformly integrable if and only if it is relatively weakly compact. From this point of view, the following theorem is a generalization of Doob Convergence Theorem to martingales in Banach lattices. Theorem 17. A relatively weakly compact martingale in a Banach lattice is fixed.

Proof. Suppose that (x_{n}) is a relatively weakly compact martingale in a Banach lattice F. By Eberlein-Šmulian Theorem it has a subsequence (x_{n_{k}}) weakly convergent to some x ∈ F. Since E_{m} is bounded and hence weakly continuous for every m, we have w-lim_{k} E_{m}x_{n_{k}} = E_{m}x, but E_{m}x_{n_{k}} = x_{m} for all sufficiently large values of k, so that x_{m} = E_{m}x. □

Corollary 18. If F is reflexive then every bounded martingale is fixed. In particular, if F is reflexive and (E_{n}) is dense then M = F.

A subset A in a Banach lattice F is said to be order bounded if A ⊆ [−x, x] for some x ∈ F_{+}, and almost order bounded if for every ε > 0 there exists x ∈ F_{+} such that A ⊆ [−x, x] + εB_{F}, where B_{F} stands for the unit ball of F. One can easily see that a subset of L_{1}(P) is uniformly integrable if and only if it is almost order bounded. If F is a Banach lattice with an order continuous norm then it follows from Theorems 10.17

12 V. G. TROITSKY

and 12.9 of [AB85] that every almost order bounded set in F is relatively weakly compact. This provides us with yet another generalization of Doob Convergence Theorem. Corollary 19. Every almost order bounded martingale in a Banach lattice with order continuous norm is fixed. Proposition 20. A weakly convergent martingale is fixed.

Proof. Suppose that (x_{n}) is a martingale such that x_{n} ⟶^{w} x. Then w-lim_{n} E_{m}x_{n} = E_{m}x for all m, but E_{m}x_{n} = x_{m} when n ⩾ m, so that x_{m} = E_{m}x. □

11. Dual filtration

Suppose that (E_{n}) is a filtration on a Banach lattice F. It is easy to see that the sequence of the adjoint operators (E_{n}^{∗}) is a filtration on the dual Banach lattice F^{∗}. Indeed, E_{n}^{∗} is contractive and positive for every n, and E_{n}^{∗}E_{m}^{∗} = (E_{m}E_{n})^{∗} = E_{n∧m}^{∗} for all n and m. In particular, E_{n}^{∗} is a projection. Proposition 21. If (E_{n}) is dense then E_{n}^{∗}f ⟶^{w^{∗}} f and ∥E_{n}^{∗}f∥ → ∥f∥ for every f ∈ F^{∗}. In particular, the map f ↦ (E_{n}^{∗}f) is an isometric embedding of F^{∗} into M(F^{∗}, (E_{n}^{∗})).

Proof. Suppose that (E_{n}) is dense. Then E_{n}x → x for each x ∈ X so that (E_{n}^{∗}f)(x) = f(E_{n}x) → f(x) for every f ∈ F^{∗}, hence E_{n}^{∗}f ⟶^{w} f. Show that ∥E_{n}^{∗}f∥ → ∥f∥. Without loss of generality, ∥f∥ = 1. Clearly, ∥E_{n}^{∗}f∥ ⩽ 1 for all n. There exists x ∈ F such that f(x) > 1 − ε. Since E_{n}x → x, there exists n such that ∥E_{n}x − x∥ < ε. It follows that |f(E_{n}x) − f(x)| < ε so that

|E_{n}^{∗}f(x) − 1| ⩽ |f(E_{n}x) − f(x)| + |f(x) − 1| < 2ε,

so that ∥E_{n}^{∗}f∥ > 1 − 2ε. It follows that ∥E_{n}^{∗}f∥ → 1. □

We also have a dual version of this result. Proposition 22. If (E_{n}^{∗}) is dense then E_{n}x ⟶^{w} x and ∥E_{n}x∥ → ∥x∥ for every x ∈ F. In particular, the map x ↦ (E_{n}x) is an isometric embedding of F into M(F, (E_{n})).

Proof. Suppose that (E_{n}^{∗}) is dense. Then E_{n}^{∗}f → f for every f ∈ F^{∗}, so that f(E_{n}x) = (E_{n}^{∗}f)(x) → f(x), for every x ∈ F, so that E_{n}x ⟶^{w} x. Fix x ∈ F and consider it as an element of F^{∗∗}, then it follows from Proposition 21 that ∥E_{n}x∥ = ∥E_{n}^{∗∗}x∥ → ∥x∥. □

The following result is the dual version of Proposition 20. Proposition 23. Every weak^{∗}-convergent martingale in F^{∗} relative to (E_{n}^{∗}) is fixed.

Proof. Suppose that (f_{n}) is a martingale in F^{∗} relative to (E_{n}^{∗}) such that f_{n} ⟶^{w^{∗}} f. Since E_{m}^{∗} is adjoint, hence weak^{∗}-continuous for each m, then E_{m}^{∗}f_{n} ⟶^{w^{∗}} E_{m}^{∗}f. But E_{m}^{∗}f_{n} = f_{m} when n ⩾ m, so that f_{m} = E_{m}^{∗}f for each m. □

Theorem 24. Given a Banach lattice F with filtration (E_{n}), denote M = M(F, (E_{n})) and Y = ⋃_{n=1}^{∞} Range E_{n}^{∗}. Then

(i) The sequence (f(x_{m})) stabilizes for every martingale (x_{m}) and every f ∈ Y. Namely, if f ∈ Range E_{n}^{∗} then f(x_{m}) = f(x_{n}) for all m ⩾ n;

(ii) Every martingale in F is σ(F, Y)-Cauchy; (iii) M embeds isometrically into Y^{∗} via X = (x_{n}) ↦ θ_{X} ∈ Y^{∗} where θ_{X}(f) = f(x_{n})

for f ∈ Range E_{n}^{∗}; (iv) If X ∈ M then X is fixed as a martingale in F^{∗∗}.

Proof. To prove (i), consider a martingale X = (x_{m}). Let f ∈ Range E_{n}^{∗} for some n. Then E_{n}^{∗}f = f, and for every m ⩾ n we have f(x_{m}) = (E_{n}^{∗}f)(x_{m}) = f(E_{n}x_{m}) = f(x_{n}). It follows also that lim_{m} f(x_{m}) exists and equals f(x_{n}). Thus, f(x_{m}) converges for every f ∈ Y, this proves (ii).

Fix a bounded martingale X = (x_{n}), for every f ∈ Range E_{n}^{∗} define θ_{X}(f) = f(x_{n}) = lim_{m} f(x_{m}). Then θ_{X} is defined for every f ∈ Y. Clearly, θ_{X} is linear. Since |f(x_{m})| ⩽ ∥f∥∥x_{m}∥ ⩽ ∥f∥∥X∥ for all m, it follows that ∥θ_{X}∥ ⩽ ∥X∥, so that θ_{X} ∈ Y^{∗}. Show that θ_{X} = ∥X∥, this will show that the map X ↦ θ_{X} is an isometry and prove (iii). Fix n, By Hahn-Banach Theorem there exists f ∈ F^{∗} such that ∥f∥ = 1 and f(x_{n}) = ∥x_{n}∥. Since E_{n}^{∗}f ∈ Range E_{n}^{∗} we have

θ_{X}(E_{n}^{∗}f) = E_{n}^{∗}f(x_{n}) = f(E_{n}x_{n}) = f(x_{n}) = ∥x_{n}∥ ⩾ ∥E_{n}^{∗}f∥∥x_{n}∥

because ∥E_{n}^{∗}f∥ ⩽ 1. Thus, ∥θ_{X}∥ ⩾ ∥x_{n}∥ for each n. It follows that ∥θ_{X}∥ = ∥X∥.

To prove (iv), extend θ_{X} to an element x^{∗∗} of F^{∗∗} by Hahn-Banach. Then for every f ∈ F^{∗} and every n we have

⟨E_{n}^{∗∗}x^{∗∗}, f⟩ = ⟨x^{∗∗}, E_{n}^{∗}f⟩ = θ_{X}(E_{n}^{∗}f) = E_{n}^{∗}f(x_{n}) = f(E_{n}x_{n}) = ⟨f, x_{n}⟩,

so that E_{n}^{∗∗}x^{∗∗} = x_{n} for every n. □

Corollary 25. If (E_{n}^{∗}) is dense then M = M(F, (E_{n})) embeds isometrically into F^{∗∗}. Namely, every martingale X = (x_{n}) in M is w^{∗}-convergent and the map Φ: X ↦ w^{*}-lim_{n} x_{n} is an isometry from M into F^{∗∗}. Also, X ⩽ Y ⇔ Φ(X) ⩽ Φ(Y)

Proof. If (E_{n}^{∗}) is dense than Y is dense in F^{∗}, so that Y^{∗} = F^{∗∗} and the first statement follows immediately from Theorem 24(iii). Given a bounded martingale X = (x_{n}) in

14 V. G. TROITSKY

F, let θ_{X} ∈ Y^{∗} be as in Theorem 24(iii). Since Y is dense in F^{∗}, we extend θ_{X} by continuity to all of F^{∗}, so that we can write θ_{X} ∈ F^{∗∗}. Show that x_{n} ⟶^{w^{∗}} θ_{X}. Given f ∈ F^{∗}, it follows from Theorem 24(i) that θ_{X}(E_{m}^{∗}f) = (E_{m}^{∗}f)(x_{m}) = f(E_{m}x_{m}) = f(x_{m}). Therefore,

|⟨x_{m}, f⟩ − ⟨θ_{X}, f⟩| = |⟨θ_{X}, E_{m}^{∗}f⟩ − ⟨θ_{X}, f⟩| ⩽ ∥θ_{X}∥∥E_{m}^{∗}f − f∥ → 0.
□

Example 26. If (E_{n}) is a filtration given by a 1-unconditional basis as in Example 1, then (e_{i}) is shrinking iff (E_{n}^{∗}) is dense, in this case M(F, (E_{n})) embeds isometrically in F^{∗∗}. Notice that in Example 2 we actually have M = F^{∗∗}.

The following can be viewd as a complement to Proposition 16.

Corollary 27. If F is a KB-space and (E_{n}^{∗}) is dense, then M(F, (E_{n})) is lattice isometric to F.

Proof. Let Φ be as in Corollary 25. Since F is weakly sequentially complete, then Φ(X) = w^{*}-lim_{n} x_{n} belongs to F for every bounded martingale X = (x_{n}). □

12. Martingales of martingales

Given a Banach lattice F and a filtration (E_{n}), let M = M(F, (E_{n})). One can view M as an enlargement or a completion of F. This enlargement operation is idempotent in the following sense. For every k ⩽ 1 consider the map E_{k} on M given as follows: if X = (x_{n}) is an element of M then E_{k}X = Y, where

y_{n} = \begin{cases}x_{n} & n ⩽ k, \\ x_{k} & n ⩾ k. \end{cases}

Clearly, Y ∈ M and ∥Y ∥ ⩽ ∥X∥, so that E_{k} is a positive contractive projection. One can easily check that E_{k}E_{m} = E_{k∧m}. Therefore, (E_{k}) is a filtration on M. Notice that if (E_{n}) preserves the norms of positive vectors, then so does (E_{k}). Indeed, if X = (x_{n}) ∈ M_{+} then ∥X∥ = ∥x_{n}∥ for all n. Let Y = E_{k}X, then Y = (y_{n}) is again a positive martingale, and ∥Y ∥ = ∥y_{1}∥ = ∥x_{1}∥ = ∥X∥.

MARTINGALES IN BANACH LATTICES 15

We claim that M(M, (E_{k})) = M up to the canonical isometry. Indeed, let (X^{(k)}) be a martingale in M(M, (E_{k})). It is easy to see that it has to be of the form

X^{(1)} = (x_{1}, x_{1}, x_{1}, x_{1}, …)
X^{(2)} = (x_{1}, x_{2}, x_{2}, x_{2}, …)
X^{(3)} = (x_{1}, x_{2}, x_{3}, x_{3}, …)
⋮

for some sequence X = (x_{n}). Then for m ⩽ n we have E_{m}x_{n} = E_{m}X_{n}^{(n)} = X_{m}^{(n)} = x_{m}, so that X is a martingale. Furthermore,

∥X∥ = sup_{n}∥x_{n}∥ = sup_{k}∥X^{(k)}∥ = ∥(X^{(k)})∥ < +∞,

so that X ∈ M. Finally, E_{k}X = X^{(k)}, so that the martingale (X^{(k)}) is fixed on X, and, clearly, X is the only element of M with this property. It follows that the map X ∈ M ↦ (E_{k}X) is a surjective isometry between M and M(M, (E_{k})).

13. Martingale difference sequences

Given a filtration (E_{n}), if m < n, then it is can be easily verified that E_{n} − E_{m} is a projection. Define the sequence (P_{n}) of difference projections via P_{1} = E_{1} and P_{n} = E_{n} − E_{n−1} for n > 1. Then E_{n} = ∑_{k=1}^{n} P_{k} for each n. Observe that P_{n}P_{m} = 0 whenever n ≠ m. Indeed,

P_{n}P_{m} = (E_{n} − E_{n−1})(E_{m} − E_{m−1}) = E_{n}E_{m} − E_{n}E_{m−1} − E_{n−1}E_{m} + E_{n−1}E_{m−1}

so that P_{n}P_{m} = E_{n} − E_{n} − E_{n−1} + E_{n−1} = 0 when n < m and P_{n}P_{m} = E_{m} − E_{m−1} − E_{m} + E_{m−1} = 0 when m < n. A sequence (d_{n}) in F is called a martingale difference sequence if d_{n} ∈ Range P_{n} for each n. Note that P_{n}d_{k} equals d_{k} if n = k and zero otherwise. If we put x_{n} = ∑_{k=1}^{n} d_{k} then one can easily verify that (x_{n}) is a martingale. Conversely, if (x_{n}) is a martingale, then the sequence (d_{n}) defined by d_{1} = x_{1} and d_{n} = x_{n} − x_{n−1} for n > 1 is a martingale difference sequence. Observe also that if (E_{n}) is dense then for every x ∈ F we have ∑_{k=1}^{n} P_{k}x = E_{n}x → x, so that x = ∑_{k=1}^{∞} P_{k}x, hence F = ⊕_{k=1}^{∞} Range P_{k}. Notice that if the ranges of P_{k} form a finite dimensional decomposition of F, then Proposition 11 guarantees that M(F, (E_{n})) is a Banach lattice.

Suppose that (d_{k}) is a martingale difference sequence. Fix a sequence of scalars (α_{k}), and put x_{n} = ∑_{k=1}^{n} α_{k}d_{k}. Clearly, X = (x_{k}) is a martingale, so that

∥∑_{k=1}^{n} α_{k}d_{k}∥ = ∥x_{n}∥ ⩽ ∥x_{n+1}∥ = ∥∑_{k=1}^{n+1} α_{k}d_{k}∥.

It follows that every martingale difference sequence is a monotone basic sequence in F.

14. Operators on martingales

Let F be a Banach lattice, and (E_{n}) a filtration on F. Let T ∈ L(F), a bounded linear operator from F to F, such that the norm limit

(2)  lim_{n→∞} E_{m}T x_{n}

exists for every bounded martingale X = (x_{n}) and every m ⩾ 1. Denote this limit by y_{m} and put Y = (y_{m}). Then Y is a martingale: if k ⩽ m then

E_{k}y_{m} = lim_{n→∞} E_{k}E_{m}T x_{n} = lim_{n→∞} E_{k}T x_{n} = y_{k}.

Notice that ∥y_{m}∥ ⩽ ∥T ∥∥X∥ for every m, so that Y is bounded and ∥Y ∥ ⩽ ∥T ∥∥X∥. It follows that T induces an operator in L(M), denote it by T̂. Clearly, ∥T̂∥ ⩽ T.

If, in addition, (E_{n}) is dense, then we know that F can be identified with the subspace of M consisting of the convergent martingales of the form (E_{n}x). In this case, T̂_{|F} = T, so that ∥T̂∥ = T

Before we discuss for which operators in L(F) the limit in (2) exists and, therefore, T̂ is defined, observe that an extension of an opeator from L(F) to L(M) need not be unique. Indeed, when F is a KB-space and (E_{n}) is dense then M = F ⊕ F^{d} by Proposition 16. If F^{d} is non-trivial, one can define the extension arbitrarily on F^{d}. Proposition 28. If T commutes with E_{n} for all n then T̂ exists.

Proof. Let X = (x_{n}) be a bounded martingale. If m ⩽ n, then E_{m}T x_{n} = T E_{m}x_{n} = T x_{m}, so that (T x_{n}) is a martingale. Put y_{n} = T x_{n} and Y = (y_{n}), then, in particular, y_{m} = lim_{n} E_{m}T x_{n} for every m, so that Y = T̂X. □

Proposition 29. If F is a KB-space and either (i) T E_{n} ⩾ E_{n}T for all n, or

(ii) T E_{n} ⩽ E_{n}T for all n,

then T̂ exists.

Proof. Let X = (x_{n}) be a bounded martingale in F. It suffices to show that the limit in (2) exists for all m. By Theorem 7, X is regular, so that we can assume without loss of generality that X ⩾ 0. Notice also that ∥E_{m}T x_{n}∥ ⩽ ∥T ∥∥X∥, so that the sequence (E_{m}T x_{n})_{n=1}^{∞} is norm bounded. If m ⩽ n then in case (i) we have

E_{m}T x_{n} = E_{m}T E_{n}x_{n+1} ⩾ E_{m}E_{n}T x_{n+1} = E_{m}T x_{n+1},

MARTINGALES IN BANACH LATTICES 17

while in case (ii) we have

E_{m}T x_{n} = E_{m}T E_{n}x_{n+1} ⩽ E_{m}E_{n}T x_{n+1} = E_{m}T x_{n+1},

In either case the sequence (E_{m}T x_{n})_{n=1}^{∞} is monotone. Since F is a KB-space, the sequence converges. □

15. Notes, remarks, and questions

The results of the present paper can be applied to vector-valued martingales based on vector measures. A beautiful exposition of vector-valued martingales can be found in [DU77]. In particular, there are deep results characterizing certain properties of a Banach space X via properties of X-valued martingales in L_{p}(Ω; X). Notice, that if F is a Banach lattice, then L_{p}(Ω; F) is again a Banach lattice, so that an F -valued martingale in the sense of [DU77] can be viewed as a martingale in L_{p}(Ω; F) in the sense of the present paper. Note that L_{p}(Ω; F) inherits many Banach lattice properties of F. In particular, one can easily verify that L_{p}(Ω; F) has order continuous norm iff F has order continuous norm, and L_{p}(Ω; F) is a KB-space iff F is a KB-space. It should be noted, however, that a vector-valued martingale which is uniformly integrable with respect to a vector measure in the sense of [DU77] need not be relatively weakly compact, so that Theorem 17 might not apply. Many interesting results about spaces L_{p}(Ω; F) of Banach lattice valued martingales can be found in [SW76, Szul78, Szul79].

R. DeMarr in [DeM66] introduced a martingale in a vector lattice F as double sequences (x_{n}, E_{n}) where x_{n} is an element of F, E_{n} is a positive linear projection, E_{n}E_{m} = E_{n∧m} and E_{n}x_{m} = x_{n} whenever n ⩽ m. DeMarr then generalized the almost everywhere part of Doob’s Convergence Theorem. Namely, he proved that under certain special conditions, a martingale in a vector lattice is order convergent.

Dor and Odell in [DO75] studied sequences of projections on L_{p}(μ) satisfying E_{n}E_{m} = E_{n∧m} and showed that such a sequence can often be reduced to a classical filtration.

The theory of non-commutative martingales is currently an active and promising subject, see [Um54, PX97, Jun02, Ran02].

We proved in this present paper that in several important special cases M(F, (E_{n})) and M_{r}(F, (E_{n})) are Banach lattices. However, we don’t have an example of a bounded non-regular martingale. Neither do we know an example where M(F, (E_{n})) is not a Banach lattice.

Let F be a Banach lattice with order continuous norm and a weak order unit. It is known (see, e.g., [LT79, Theorem 1.b.14]) that F is order isometric to a norm dense ideal

18 V. G. TROITSKY

of L_{1}(P) for some probability space (Ω, F, P). It would be interesting to know if this representation could be chosen in such a way that a sequence in F is a martingale if and only if its image in L_{1}(P) is a martingale in the classical sense?

In the classical case, given a martingale X = (x_{n}) in L_{1}(P), one can recover a filtration with respect to which X is a martingale. Namely, one can take F_{n} to be the smallest σ-algebra that makes x_{1}, …, x_{n} measurable. In this approach a martingale is a primary object, while the filtration is recovered from the martingale. It is interesting if there is a similar procedure in the Banach lattice case. That is, given a sequence (x_{n}) of elements of a Banach lattice is a martingale, when can one find a filtration (E_{n}) with respect to which (x_{n}) is a martingale?

There are several important martingale inequalities in the classical martingale theory involving maximal function, martingale transform, subordinate martingales, and escape numbers, see [BDG72, Bur81, Bur84, Bur01]. It would be interesting to find analogues of these inequalities in Banach lattice setting.

In the present paper we considered spaces of martingales relative to a fixed filtration. It would seem natural to consider the set of all martingales corresponding to all filtrations on a given Banach lattice and to try to establish some order structure on this set. However, there are certain difficulties on the way, even in the classical setting. For example, let X = (x_{n}) be the double-or-nothing martingale on the unit interval, and let Y = (y_{n}) where y_{n} = x_{n+1}, so that Y is just a “shift” of X. Then X and Y are not only distinct, but even non-comparable in M_{1}. We don’t see any reasonable way to define the X ∨ Y.

In this paper we only dealt with discrete martingales, ordered by positive integers. Most of the results can be easily generalized to continuous martingales.

The author would like to thank D. Burkholder, E. Emel’yanov, and R. Vershynin for their interest in the work and numerous suggestions. Thanks are also due to the referee for helpful remarks. The author is grateful to the University of Alberta for hospitality and support. This paper would not have existed without inspiring discussions with Yuri Abramovich, my friend and advisor.

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Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1. Canada.

  E-mail address: vtroitsky@math.ualberta.ca