▁such ▁al gebras ▁are ▁rarely ▁separ able ▁in ▁the ▁norm ▁topology . ▁The ▁von ▁Ne umann ▁algebra ▁generated ▁by ▁a ▁set ▁of ▁bounded ▁operators ▁on ▁a ▁Hil bert ▁space ▁is ▁the ▁smallest ▁von ▁Ne umann ▁algebra ▁containing ▁all ▁those ▁operators . ▁The ▁tensor ▁product ▁of ▁two ▁von ▁Ne umann ▁al gebras ▁acting ▁on ▁two ▁Hil bert ▁spaces ▁is ▁defined ▁to ▁be ▁the ▁von ▁Ne umann ▁algebra ▁generated ▁by ▁their ▁algebraic ▁tensor ▁product , ▁considered ▁as ▁operators ▁on ▁the ▁Hil bert ▁space ▁tensor ▁product ▁of ▁the ▁Hil bert ▁spaces . ▁ ▁By ▁forget ting ▁about ▁the ▁topology ▁on ▁a ▁von ▁Ne umann ▁algebra , ▁we ▁can ▁consider ▁it ▁a ▁( un ital ) ▁* - algebra , ▁or ▁just ▁a ▁ring . ▁Von ▁Ne umann ▁al gebras ▁are ▁semi her edit ary : ▁every ▁finit ely ▁generated ▁sub module ▁of ▁a ▁project ive ▁module ▁is ▁itself ▁project ive . ▁There ▁have ▁been ▁several ▁attempts ▁to ▁ax iom at ize ▁the ▁underlying ▁rings ▁of ▁von ▁Ne umann ▁al gebras , ▁including ▁Ba er ▁* - r ings ▁and ▁A W * - al gebras . ▁The ▁* - algebra ▁of ▁affili ated ▁operators ▁of ▁a ▁finite ▁von ▁Ne umann ▁algebra ▁is ▁a ▁von ▁Ne umann ▁regular ▁ring . ▁( The ▁von ▁Ne umann ▁algebra ▁itself ▁is ▁in ▁general ▁not ▁von ▁Ne umann ▁regular .) ▁ ▁Comm ut ative ▁von ▁Ne umann ▁al gebras ▁▁ ▁The ▁relationship ▁between ▁commut ative ▁von ▁Ne umann ▁al gebras ▁and ▁measure ▁spaces ▁is ▁analog ous ▁to ▁that ▁between ▁commut ative