▁is omorphic ▁to ▁M ∗ . ▁ ▁showed ▁that ▁the ▁existence ▁of ▁a ▁pre du al ▁character izes ▁von ▁Ne umann ▁al gebras ▁among ▁C * ▁al gebras . ▁ ▁The ▁definition ▁of ▁the ▁pre du al ▁given ▁above ▁seems ▁to ▁depend ▁on ▁the ▁choice ▁of ▁Hil bert ▁space ▁that ▁M ▁acts ▁on , ▁as ▁this ▁determ ines ▁the ▁ult ra weak ▁topology . ▁However ▁the ▁pre du al ▁can ▁also ▁be ▁defined ▁without ▁using ▁the ▁Hil bert ▁space ▁that ▁M ▁acts ▁on , ▁by ▁defining ▁it ▁to ▁be ▁the ▁space ▁generated ▁by ▁ ▁all ▁positive ▁normal ▁linear ▁function als ▁on ▁M . ▁ ▁( Here ▁" normal " ▁means ▁that ▁it ▁pres erves ▁supre ma ▁when ▁applied ▁to ▁increasing ▁n ets ▁of ▁self ▁ad joint ▁operators ; ▁or ▁equival ently ▁to ▁increasing ▁sequences ▁of ▁pro jections .) ▁ ▁The ▁pre du al ▁M ∗ ▁is ▁a ▁closed ▁sub space ▁of ▁the ▁dual ▁M * ▁( which ▁consists ▁of ▁all ▁norm - continu ous ▁linear ▁function als ▁on ▁M ) ▁but ▁is ▁generally ▁smaller . ▁The ▁proof ▁that ▁M ∗ ▁is ▁( us ually ) ▁not ▁the ▁same ▁as ▁M * ▁is ▁non construct ive ▁and ▁uses ▁the ▁ax iom ▁of ▁choice ▁in ▁an ▁essential ▁way ; ▁it ▁is ▁very ▁hard ▁to ▁exhib it ▁explicit ▁elements ▁of ▁ ▁M * ▁that ▁are ▁not ▁in ▁M ∗ . ▁For ▁example , ▁ex otic ▁positive ▁linear ▁forms ▁on ▁the ▁von ▁Ne umann ▁algebra ▁l ∞ ( Z ) ▁are ▁given ▁by ▁free ▁ult ra filters