Lipecki, Zbigniew Order intervals in Banach lattices and their extreme points. (English) Zbl 1459.46023 Colloq. Math. 160, No. 1, 119-132 (2020). 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 \(x^*\in X_+^*\)\) Cited in 3 Documents 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) Keywords:linear lattice; weak order unit; order interval; extreme point; atom; atomic; nonatomic; locally solid; Banach lattice; closed; weakly closed; weakly dense; weak\(^\ast\) closed; weak\(^\ast\) dense; order continuous; Boolean algebra; Lyapunov’s convexity theorem PDFBibTeX XMLCite \textit{Z. Lipecki}, Colloq. Math. 160, No. 1, 119--132 (2020; Zbl 1459.46023) Full Text: DOI