MARTINGALES IN BANACH LATTICES, II

HAILEGEBRIEL E. GESSESSE AND VLADIMIR G. TROITSKY

Abstract. This note is a follow-up to [Tro05]. We provide several sufficient conditions for the space M of bounded martingale on a Banach lattice F to be a Banach lattice itself. We also present examples in which M is not a Banach lattice. It is shown that if F is a KB-space and the filtration is dense then F is a projection band in M.

Introduction

This short note is a follow-up to [Tro05], where the second author introduced and studied spaces of bounded martingales on Banach lattices. Let us briefly recall some key definitions from [Tro05]. Throughout this paper, F is a Banach lattice. By a filtration on F we mean a sequence (E_{n}) of positive contractive projections such that E_{n}E_{m} = E_{n∧m}. A sequence (x_{n}) in F is said to be a martingale (a submartingale) relative to a filtration (E_{n}) if E_{n}x_{m} = x_{n} (E_{n}x_{m} ⩾ x_{n}, respectively) whenever n ⩽ m. A (sub)martingale X = (x_{n}) is bounded if it has finite martingale norm given by ∥X∥ = sup_{n}∥x_{n}∥. We write M = M(F, (E_{n})) for the space of all bounded martingale on F relative to filtration (E_{n}). It is easy to see that M is a Banach space. Also, M can be ordered component-wise, i.e., (x_{n}) ⩽ (y_{n}) if x_{n} ⩽ y_{n} for every n. It is easy to see that, under this order, M is an ordered Banach space and the norm is monotone, i.e., 0 ⩽ X ⩽ Y implies ∥X∥ ⩽ ∥Y ∥. It was shown in [Tro05] that under certain conditions on F the space M is itself a Banach lattice. In Section 1 of this note, we slightly improve some of these conditions. However, the question whether M is always a Banach lattice was left unanswered in [Tro05]. In Section 2 of this note, we answer this question in the negative by providing examples in which M is not a Banach lattice.

Section 3 is concerned with the case when F is a KB-space. It was shown in [Tro05] that in this case, M is a Banach lattice. It was also claimed in [Tro05] that in this case, F can be identified with a projection band in M. However, the proof of the latter claim in [Tro05] contained a gap. In Section 3 of this note, we present a complete proof of the assertion.

Positivity, to appear.

1 2 H. E. GESSESSE AND V. G. TROITSKY

Date: December 18, 2009.

2000 Mathematics Subject Classification. 60G48, 46B42.

Key words and phrases. Martingale, filtration, Banach lattice.

1. When is M a Banach lattice?

We start by extending Lemmas 5 and 6 of [Tro05] to weakly convergent sequences. The proofs are analogous.

Lemma 1. 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 weakly to some (z_{n}) for each n, then Z = (z_{n}) is a bounded martingale, and it is the least martingale satisfying X ⩽ Z and Y ⩽ Z.

Proof. (i) Let 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 ∥.

(ii) Suppose that w-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}(w-lim_{m→∞} E_{n}(x_{m} ∨ y_{m})) = w-lim_{m→∞} E_{k}E_{n}(x_{m} ∨ y_{m}) = w-lim_{m→∞} E_{k}(x_{m} ∨ y_{m}) = z_{k}.

Furthermore, by properties of weak convergence, we have

∥z_{n}∥ ⩽ lim inf_{m→∞}∥E_{n}(x_{m} ∨ y_{m})∥ ⩽ ∥X∥ + ∥Y ∥

for every n, so that Z is bounded. 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 w-lim_{m} E_{n}(x_{m} ∨ y_{m}) = z_{n}, this yields z̃_{n} ⩾ z_{n}, so that Z̃ ⩾ Z. □

Corollary 2. Suppose that w-lim_{m} E_{n}|x_{m}| exists for each n and for each martingale (x_{n}) in M. Then M is a Banach lattice with lattice operations given by (X ∨ Y)_{n} =

w-lim_{m→∞} E_{n}(x_{m} ∨ y_{m}), |X|_{n} = w-lim_{m→∞} E_{n}|x_{m}|, etc, for any X, Y ∈ M with X = (x_{n}) and

Y = (y_{n}).

Proof. Let X, Y ∈ M, put Z = X − Y, then Z ∈ M. Write X = (x_{n}), Y = (y_{n}), and Z = (z_{n}). Then for n ⩽ m we have

E_{n}(x_{m} ∨ y_{m}) = E_{n}(\frac{x_{m} + y_{m}}{2} + \frac{|x_{m} − y_{m}|}{2}) = \frac{x_{n} + y_{n}}{2} + \frac{1}{2}E_{n}|z_{m}|,

MARTINGALES IN BANACH LATTICES, II 3

which converges weakly as m → +∞ by the hypothesis. Thus, by Lemma 1, X ∨ Y is a bounded martingale. Hence, M is a vector lattice with lattice operation as in the statement.

It remains to show that ∥|X|∥ = ∥X∥ for every X ∈ M. Let Z = |X|. Write X = (x_{n}) and Z = (z_{n}). Then z_{n} = w-lim_{m} E_{n}|x_{m}| for every n. Let U = {f ∈ F_{+}^{∗} : ∥f∥ ⩽ 1}. Then

(1)  ∥z_{n}∥ = sup_{f∈U} f(z_{n}) = sup_{f∈U} lim_{m→∞} f(E_{n}|x_{m}|).

Note that for every f ∈ U we have f(E_{n}|x_{m}|) ⩽ ∥f∥∥E_{n}∥∥x_{m}∥ ⩽ ∥X∥, so that ∥Z∥ ⩽ ∥X∥. On the other hand, for n ⩽ m we have |x_{n}| = |E_{n}x_{m}| ⩽ E_{n}|x_{m}|, so that f(E_{n}|x_{m}|) ⩾ f(|x_{n}|). It follows from (1) that ∥z_{n}∥ ⩾ ∥|x_{n}|∥ = ∥x_{n}∥, hence ∥Z∥ ⩾ ∥X∥. □

It was shown in [Tro05] that if F is a KB-space then M is a Banach lattice. We can now prove the following stronger result.

Corollary 3. Suppose that every increasing norm bounded sequence in F converges weakly. Then M is a Banach lattice. If X, Y ∈ M with X = (x_{n}) and Y = (y_{n})

then (X ∨ Y)_{n} = w-lim_{m→∞} E_{n}(x_{m} ∨ y_{m}).

Proposition 4. If E_{n} is a band projection for every n then M is a Banach lattice with coordinate-wise lattice operations.

Proof. Let X = (x_{n}) ∈ M. Then for every m ⩾ n we have E_{n}|x_{m}| = |E_{n}x_{m}| = |x_{n}|. The conclusion now follows from Corollary 2. □

Theorem 5. If F is order continuous, then the following statments are equivalent. (i) M is a Banach lattice. (ii) For each n, (E_{n}|x_{m}|)_{m} converges weakly for each (x_{n}) ∈ M. (iii) For each n, (E_{n}|x_{m}|)_{m} converges in norm for each (x_{n}) ∈ M.

Proof. (i)⇒(ii) Suppose M is a Banach lattice and let X = (x_{n}) ∈ M. Then |X| exists in M, say, |X| = (y_{n}) ∈ M. It follows from |x_{m}| ⩽ y_{m} that 0 ⩽ E_{n}|x_{m}| ⩽ E_{n}y_{m} = y_{n} whenever n ⩽ m. Thus, the sequence (E_{n}|x_{m}|)_{m=n}^{∞} is increasing and bounded above. Since F is order continuous, order interval in F are weakly compact, see, e.g. [AB85, Theorem 12.9]. Hence (E_{n}|x_{m}|) has a weakly convergent subsequence. It follows from Lemma 1(i) that (E_{n}|x_{m}|) is increasing, hence the entire sequence converges weakly.

(ii)⇒(iii) By Lemma 1(i), (E_{n}|x_{m}|)_{m} is increasing, and every increasing weakly convergent sequence is norm convergent (see, e.g., Proposition 1.4.1 of [MN91].

(iii)⇒(i) This is just a special case of Corollary 2. □

2. Examples when M is not a Banach lattice

Example 6. In this example we construct a filtration (E_{n}) on c_{0} such that M(c_{0}, (E_{n})) is not a Banach lattice.

As usually, an operator T ∈ L(c_{0}) can be represented by an infinite matrix where the j-column is T e_{j}. For n = 0, 1, 2, …, put

E_{n} = [\begin{matrix}1 \\ & . & . & . \\ & & & & 1 \\ & & & & & 1/2 & 1/2 \\ & & & & & 1/2 & 1/2 \\ & & & & & & & 1/2 & 1/2 \\ & & & & & & & 1/2 & 1/2 \\ & & & & & & & & & . & . & . \end{matrix} ]

with 2n ones in the upper-left corner. In other words,

E_{n}e_{i} = e_{i} if i ⩽ 2n, and E_{n}e_{2k−1} = E_{n}e_{2k} = \frac{1}{2}(e_{2k−1} + e_{2k}) when n < k.

Note that E_{0} has no 1’s at all. It is easy to see that (E_{n}) is a dense filtration. Furthermore, for n = 0, 1, 2, …, put

x_{n} = (\underbrace{−1, 1, …, −1, 1}_{2n}, 0, …).

It is easy to see that (x_{n}) is a bounded martingale relative to (E_{n}). On the other hand,

E_{0}|x_{n}| = (\underbrace{1, 1, …, 1, 1}_{2n}, 0, …).

Clearly, E_{0}|x_{n}| diverges, so that M is not a Banach lattice by Theorem 5.

Example 7. This is another example where M is not a Banach lattice. For the definition and properties of vector-valued L_{p}-spaces we refer the reader to [DU77]. Suppose that X is a Banach space, and let a sequence (a_{n})_{n=0}^{∞} in X be a tree, that is, a_{n} = \frac{1}{2}(a_{2n+1}+a_{2n+2}) for every n. Now define a sequence (x_{n}) in L_{1}([0, 1], X) via

x_{0} = a_{0}χ_{[0,1]},
x_{1} = a_{1}χ_{[0, \frac{1}{2} )} + a_{2}χ_{[ \frac{1}{2} ,1]},
x_{2} = a_{3}χ_{[0, \frac{1}{4} )} + a_{4}χ_{[ \frac{1}{4} , \frac{2}{4} )} + a_{5}χ_{[ \frac{2}{4} , \frac{3}{4} )} + a_{6}χ_{[ \frac{3}{4} ,1]},

etc.

Then (x_{n}) is a martingale in the sense of [DU77] relative to the dyadic filtration of [0, 1].

MARTINGALES IN BANACH LATTICES, II 5

Now, suppose that X is a Banach lattice. Then L_{1}([0, 1], X) also is a Banach lattice. Next, we define a sequence of projections on L_{1}([0, 1], X) as follows. For f ∈ L_{1}([0, 1], X), we put

E_{0}f = (∫_{0}^{1} f)χ_{[0,1]},
E_{1}f = 2[(∫_{0}^{1/2} f)χ_{[0, \frac{1}{2} )} + (∫_{1/2}^{1} f)χ_{[ \frac{1}{2} ,1]}],
E_{2}f = 4[(∫_{0}^{1/4} f)χ_{[0, \frac{1}{4} )} + (∫_{1/4}^{2/4} f)χ_{[ \frac{1}{4} , \frac{2}{4} )} + (∫_{2/4}^{3/4} f)χ_{[ \frac{2}{4} , \frac{3}{4} )} + (∫_{3/4}^{1} f)χ_{[ \frac{3}{4} ,1]}],

etc.

It is easy to see that (E_{n}) is a filtration on L_{1}([0, 1], X), and (x_{n}) is a martingale in L_{1}([0, 1], X) relative to this filtration.

Now put X = c_{0} and let F = L_{1}([0, 1], c_{0}). Let (x_{k}) be defined as above with

a_{0} = (0, …),
a_{1} = (1, 0, …), a_{2} = (−1, 0, …),

a_{3} = (1, 1, 0, …), a_{4} = (1, −1, 0 …), a_{5} = (−1, 1, 0, …), a_{6} = (−1, −1, 0, …),

etc. In other words, x_{n} = (r_{1}, …, r_{n}, 0, …), where r_{k} is the k-th Rademacher function. Then ∥x_{n}∥ = 1 for every n. However,

E_{0}|x_{n}| = (\underbrace{𝟙, …, 𝟙}_{n times}, 0 …).

Again, E_{0}|x_{n}| diverges, so that M is not a Banach lattice by Theorem 5.

3. How does F sit in M?

Again, throughout this section we assume that F is a Banach lattice, (E_{n}) is a filtration on F, and M = M(F, (E_{n})). Moreover, we will assume that (E_{n}) is dense, i.e., E_{n}x → x for every x ∈ F. It was observed in Section 8 of [Tro05] that in this case a bounded martingale (x_{n}) converges iff it is fixed, i.e., there exists x ∈ F such that x_{n} = E_{n}x for every n. Clearly, in this case we have x_{n} → x.

Define φ: F → M via φ(x) = (E_{n}x)_{n=1}^{∞}. It is clear that φ is an isometry. We claim that φ is a lattice homomorphism, so that F is lattice isometric to a closed subspace of M.

Indeed, take any x, y ∈ F and put x_{n} = E_{n}x and y_{n} = E_{n}y for all n. Since the lattice operations are continuous, we have x_{n} ∨ y_{n} → x ∨ y. Hence, for every n we have

lim_{m→∞} E_{n}(x_{m} ∨ y_{m}) = E_{n}( lim_{m→∞} x_{m} ∨ y_{m}) = E_{n}(x ∨ y).

It follows from Lemma 1 that φ(x) ∨ φ(y) = φ(x ∨ y). Finally,

φ(x ∧ y) = −φ((−x) ∨ (−y)) = −(φ(−x) ∨ φ(−y)) = φ(x) ∧ φ(y).

Lemma 8. Suppose that F is order continuous and M is a Banach lattice. Then M is order complete. If, in addition, (E_{n}) is dense, then φ(F) is an ideal in M.

Proof. First, show that M is order complete. Suppose that 0 ⩽ X^{(α)} ↑⩽ X in M. Put X = (x_{n}) and X^{(α)} = (x_{n}^{(α)}). Then 0 ⩽ x_{n}^{(α)} ↑⩽ x_{n} for every n. Since F is order continuous, for every n there exists y_{n} such that we have x_{n}^{(α)} → y_{n}. Put Y = (y_{n}). It is easy to see that Y is a martingale. It follows from 0 ⩽ Y ⩽ X that Y is bounded, hence Y ∈ M.

Now suppose that (E_{n}) is dense. Put M_{0} = φ(F). Then F is lattice isometric to M_{0}. Show that M_{0} is an ideal in M. Suppose that 0 ⩽ X ⩽ Y for some X ∈ M and Y ∈ M_{0}. Put X = (x_{n}) and Y = (y_{n}). Then 0 ⩽ x_{n} ⩽ y_{n} for every n and there exists y ∈ F such that y_{n} = E_{n}y for all n. Fix ε > 0. It follows from y_{n} → y that there exists n_{0} such that ∥y_{n} − y∥ < ε whenever n ⩾ n_{0}. It follows that

|x_{n} − x_{n} ∧ y| = |x_{n} ∧ y_{n} − x_{n} ∧ y| ⩽ |y_{n} − y|,

so that ∥x_{n} − x_{n} ∧ y∥ < ε. If follows from x_{n} ∧ y ∈ [0, y] that x_{n} ∈ [0, y] + B_{ε} for all n ⩾ n_{0}. Therefore, (x_{n}) is almost order bounded. Hence, it converges by Corollary 19 of [Tro05]. Hence, X ∈ M_{0}. Now suppose that Y ∈ M_{0} and X ∈ M such that |X| ⩽ Y. Then 0 ⩽ X^{+}, X^{−} ⩽ Y, so that X^{+}, X^{−} ∈ M_{0} and, therefore, X ∈ M_{0}. Thus, M_{0} is an ideal. □

Now we are ready to present a new proof of Proposition 16 of [Tro05].

Theorem 9. If F is a KB-space and (E_{n}) is dense then φ(F) is a projection band in M.

Proof. By Theorems 7 of [Tro05] and Lemma 8, M is an order complete Banach lattice. Again, denote M_{0} = φ(F). By Lemma 8, M_{0} is an ideal in M. It is left to show that M_{0} is a band because every band in an order complete lattice is a projection band by Theorems 3.8 of [AB85].

To show that M_{0} is a band, suppose that 0 ⩽ X^{(α)} ↑ X for some net (X^{(α)}) in M_{0} and some X ∈ M. Put X = (x_{n}) and X^{(α)} = (x_{n}^{α}). Let X^{(α)} = φ(x^{(α)}) for some x^{(α)} ∈ F. Clearly, ∥x^{(α)}∥ = ∥X^{(α)}∥ ⩽ ∥X∥ for every α, hence the net (x^{(α)}) is norm bounded. Since F is a KB-space, this net converges in norm to some y ∈ F, see [AB85, p. 225]. It follows also that x^{(α)} ↑ y in F. Put Y = φ(y), Y = (y_{n}). For every α we have x^{(α)} ⩽ y, so that X^{(α)} ⩽ Y, hence X ⩽ Y. On the other hand, x^{(α)} → y implies lim_{α} x_{n}^{(α)} = y_{n} for every n. Together with x_{n}^{(α)} ⩽ x_{n} this implies y_{n} ⩽ x_{n}, so that Y ⩽ X. Thus, X = Y, so that X ∈ M_{0}. □

References

[AB85] C. D. Aliprantis and O. Burkinshaw, Positive operators. Academic Press Inc., Orlando, Fla., 1985.

[CL07] S. F. Cullender and C. C. A Labuschagne, Unconditional Schauder decompositions and stopping times in the Lebesgue-Bochner spaces. J. Math. Anal. Appl., 336 (2007), no. 2, 849–864.

[DU77] J. Diestel and J. J. Uhl, Jr., Vector measures. American Mathematical Society, Providence, R.I., 1977. Mathematical Surveys, No. 15.

[MN91] P. Meyer-Nieberg, Banach lattices. Springer-Verlag, Berlin, 1991.

[Tro05] V.G. Troitsky, Martingales in Banach lattices, Positivity, no. 9(3), 2005, 437–456.

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

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