Extreme points of Banach lattices related to conditional expectations |
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Authors: | Pei-Kee Lin |
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Institution: | Department of Mathematics, University of Memphis, Memphis, TN 38152, USA |
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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?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
is a convex combination of at most 2n extreme points in the closed unit ball of K.
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Keywords: | Conditional expectation Extreme point Banach lattice Uniformly λ-property |
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