Extreme points of Banach lattices related to conditional expectations

Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f ∈ L1(X,F,μ): Φ(|f |) ∞ < ∞} with the norm f = Φ(|f |) ∞. We prove the following theorems: (1) The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE)) = X. (2) Suppose that there is n ∈ N such that f nΦ(f ) for all positive f in L∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with f 1 n is a convex combination of at most 2n extreme points in the closed unit ball of K.  2005 Elsevier Inc. All rights reserved