## Order intervals in Banach lattices and their extreme points

### Volume 160 / 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 $x^*\in X_+^*$.