ORDER INTERVALS IN BANACH LATTICES AND THEIR EXTREME POINTS

Let X be a Banach lattice with order continuous norm. Then (A) X is atomic if and only if extr[0, x] is weakly closed for every x is an element of X+ if and only if the weak and strong topologies coincide on [0, x] for every x is an element of X+ ; (B) X is nonatomic if and only if extr[0, x] is weakly dense in [0, x] for every x is an element of X+. Let, in addition, X have a weak order unit. Then (C) X* is atomic if and only if extr[0, x*] is weak* closed for every x is an element of X*(+); (D) X* is nonatomic if and only if extr[0, x*] is weak* dense in [0, x*] for every x is an element of X*(+).