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‖⩽1n is a convex combination of at most 2n extreme points in the closed unit ball of K.