Extreme points of Banach lattices related to conditional expectations
Abstract
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
Recommended Citation
Lin, P. (2005). Extreme points of Banach lattices related to conditional expectations. Journal of Mathematical Analysis and Applications, 312 (1), 138-147. https://doi.org/10.1016/j.jmaa.2005.03.024
