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 les; nΦ(f) for all positive f in L∞ 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.

Publication Title

Journal of Mathematical Analysis and Applications