; ▁they ▁correspond ▁to ▁ex otic ▁* - hom omorph isms ▁into ▁C ▁and ▁describe ▁the ▁Stone – Č ech ▁compact ification ▁of ▁Z . ▁ ▁Ex amples : ▁The ▁pre du al ▁of ▁the ▁von ▁Ne umann ▁algebra ▁L ∞ ( R ) ▁of ▁essentially ▁bounded ▁functions ▁on ▁R ▁is ▁the ▁Ban ach ▁space ▁L 1 ( R ) ▁of ▁integra ble ▁functions . ▁The ▁dual ▁of ▁L ∞ ( R ) ▁is ▁strictly ▁larger ▁than ▁L 1 ( R ) ▁For ▁example , ▁a ▁functional ▁on ▁L ∞ ( R ) ▁that ▁extends ▁the ▁Di rac ▁measure ▁ δ 0 ▁on ▁the ▁closed ▁sub space ▁of ▁bounded ▁continuous ▁functions ▁C 0 b ( R ) ▁cannot ▁be ▁represented ▁as ▁a ▁function ▁in ▁L 1 ( R ). ▁The ▁pre du al ▁of ▁the ▁von ▁Ne umann ▁algebra ▁B ( H ) ▁of ▁bounded ▁operators ▁on ▁a ▁Hil bert ▁space ▁H ▁is ▁the ▁Ban ach ▁space ▁of ▁all ▁trace ▁class ▁operators ▁with ▁the ▁trace ▁norm ▁|| A || = ▁Tr (| A | ). ▁The ▁Ban ach ▁space ▁of ▁trace ▁class ▁operators ▁is ▁itself ▁the ▁dual ▁of ▁the ▁C * - algebra ▁of ▁compact ▁operators ▁( which ▁is ▁not ▁a ▁von ▁Ne umann ▁algebra ). ▁ ▁We ights , ▁states , ▁and ▁traces ▁We ights ▁and ▁their ▁special ▁cases ▁states ▁and ▁traces ▁are ▁discussed ▁in ▁detail ▁in ▁. ▁ ▁A ▁weight ▁ ω ▁on ▁a ▁von ▁Ne umann ▁algebra ▁is ▁a ▁linear ▁map ▁from ▁the ▁set ▁of ▁positive ▁elements ▁( th ose