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Order intervals in Banach lattices and their extreme points. (English) Zbl 1459.46023

Summary: Let \(X\) be a Banach lattice with order continuous norm. Then (A) \(X\) is atomic if and only if \(\operatorname{extr} [0,x]\) is weakly closed for every \(x\in X_+\) if and only if the weak and strong topologies coincide on \([0,x]\) for every \(x\in X_+$; (B)~$ X\) is nonatomic if and only if \(\operatorname{extr} [0,x]\) is weakly dense in \([0,x]\) for every \(x\in X_+\). Let, in addition, \(X\) have a weak order unit. Then (C) \( X^*\) is atomic if and only if \(\operatorname{extr} [0,x^*]\) is weak\)^*\) closed for every \(x^*\in X_+^*\); (D) \(X^*\) is nonatomic if and only if \(\operatorname{extr} [0,x^*]\) is weak\(^*\) dense in \([0,x^*]\) for every \(x^*\in X_+^*\)\)

MSC:

46B42 Banach lattices
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
28B05 Vector-valued set functions, measures and integrals
06E99 Boolean algebras (Boolean rings)
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