A Geometrical Characterization of Banach Spaces in Which Martingale Difference Sequences are Unconditional

Abstract

Let 1 < p < ∞. For what Banach spaces B does there exist a positive real number c _p such that

$$||{\varepsilon _1}{d_1} + \cdots + {\varepsilon _n}{d_1}|{|_p} \leq {c_p}||{d_1} +\cdots + {d_n}|{|_p}$$

for all B-valued martingale difference sequences d = (d₁, d₂,…) all numbers є₁є₂,… in {-1, 1}, and all n ≥ 1? This and closely related questions have been of interest to Maurey [16], Pisier [17], Diestel and Uhl [12], Aldous [1], and others. Let us write B∈E UMD (the space B has the unconditionality property for martingale differences) if such a constant Cp = Cp(B) does exist. (Maurey uses a slightly different notation.) This class of spaces appears to depend on p but, in fact, does not [16] as we shall see in another way. It was proved in [4] that ℝ ∈ UMD and from this follows immediately that the Lebesgue spaces l^r, L^r(0 1) E UMD for 1 < r < ∞. Any UMD-space is reflexive, in fact superreflexive [16], [1], so, for example, l¹,l^∞E UMD. On the other hand, Pisier [17] has constructed an example showing that a suoerreflexive space need not be UMD.