▁The ▁concrete ▁and ▁abstract ▁definitions ▁of ▁a ▁von ▁Ne umann ▁algebra ▁are ▁similar ▁to ▁the ▁concrete ▁and ▁abstract ▁definitions ▁of ▁a ▁C * - algebra , ▁which ▁can ▁be ▁defined ▁either ▁as ▁norm - closed ▁* - al gebras ▁of ▁operators ▁on ▁a ▁Hil bert ▁space , ▁or ▁as ▁Ban ach ▁* - al gebras ▁such ▁that ▁|| aa * || = || a || ▁|| a * || . ▁ ▁Term in ology ▁Some ▁of ▁the ▁termin ology ▁in ▁von ▁Ne umann ▁algebra ▁theory ▁can ▁be ▁confusing , ▁and ▁the ▁terms ▁often ▁have ▁different ▁mean ings ▁outside ▁the ▁subject . ▁ ▁A ▁factor ▁is ▁a ▁von ▁Ne umann ▁algebra ▁with ▁trivial ▁center , ▁i . e . ▁a ▁center ▁consisting ▁only ▁of ▁scalar ▁operators . ▁A ▁finite ▁von ▁Ne umann ▁algebra ▁is ▁one ▁which ▁is ▁the ▁direct ▁integral ▁of ▁finite ▁factors ▁( mean ing ▁the ▁von ▁Ne umann ▁algebra ▁has ▁a ▁faithful ▁normal ▁tra cial ▁state ▁ τ : ▁M ▁→ ℂ , ▁see ▁http :// pers o . ens - ly on . fr / g abor ia u / even ements / I HP - trim ester / I HP - C IR M / Not es = Cy ril = finite - von Ne umann . pdf ). ▁ ▁Similarly , ▁properly ▁infinite ▁von ▁Ne umann ▁al gebras ▁are ▁the ▁direct ▁integral ▁of ▁properly ▁infinite ▁factors . ▁A ▁von ▁Ne umann ▁algebra ▁that ▁acts ▁on ▁a ▁separ able ▁Hil bert ▁space ▁is ▁called ▁separ able . ▁ ▁Note ▁that