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 ∈ L 1 ( 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 ( χ E ) ) = X . (2) Suppose that there is n ∈ N such that f ⩽ n Φ ( 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 2 n extreme points in the closed unit ball of K .