Order intervals in Banach lattices and their extreme points
Volume 160 / 2020
Colloquium Mathematicum 160 (2020), 119-132
MSC: 46A40, 46B42, 52A07, 28B05, 06E99.
DOI: 10.4064/cm7726-5-2019
Published online: 15 January 2020
Abstract
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 $x^*\in X_+^*$.
