Extreme points of Banach lattices related to conditional expectations

Let (X, F, mu) be a complete probability space, beta a sub-sigma-algebra, and phi the probabilistic conditional expectation operator determined by beta. Let kappa be the Banach lattice {f is an element of L-1 (X, F, mu): parallel to phi(vertical bar f vertical bar)parallel to(infinity) < infinity} with the norm parallel to f parallel to = parallel to phi(vertical bar f vertical bar)parallel to(infinity). We prove the following theorems: (1) The closed unit ball of kappa contains an extreme point if and only if there is a localizing set E for beta such that supp (phi (chi(E))) = X. (2) Suppose that there is n is an element of N such that f <= n phi(f) for all positive f in L-infinity(X,.F, mu). Then kappa has the uniformly lambda-property and every element f in the complex kappa with parallel to f parallel to <= 1/n is a convex combination of at most 2n extreme points in the closed unit ball of kappa. (c) 2005 Elsevier Inc. All rights reserved.