- al gebras , ▁so ▁in ▁particular ▁any ▁von ▁Ne umann ▁algebra ▁is ▁a ▁C * - algebra . ▁ ▁The ▁second ▁definition ▁is ▁that ▁a ▁von ▁Ne umann ▁algebra ▁is ▁a ▁subset ▁of ▁the ▁bounded ▁operators ▁closed ▁under ▁in volution ▁( the ▁* - operation ) ▁and ▁equal ▁to ▁its ▁double ▁commut ant , ▁or ▁equival ently ▁the ▁commut ant ▁of ▁some ▁subset ▁closed ▁under ▁*. ▁The ▁von ▁Ne umann ▁double ▁commut ant ▁theorem ▁ ▁says ▁that ▁the ▁first ▁two ▁definitions ▁are ▁equivalent . ▁ ▁The ▁first ▁two ▁definitions ▁describe ▁a ▁von ▁Ne umann ▁algebra ▁con cret ely ▁as ▁a ▁set ▁of ▁operators ▁acting ▁on ▁some ▁given ▁Hil bert ▁space . ▁ ▁showed ▁that ▁von ▁Ne umann ▁al gebras ▁can ▁also ▁be ▁defined ▁abstract ly ▁as ▁C * - al gebras ▁that ▁have ▁a ▁pre du al ; ▁in ▁other ▁words ▁the ▁von ▁Ne umann ▁algebra , ▁considered ▁as ▁a ▁Ban ach ▁space , ▁is ▁the ▁dual ▁of ▁some ▁other ▁Ban ach ▁space ▁called ▁the ▁pre du al . ▁ ▁The ▁pre du al ▁of ▁a ▁von ▁Ne umann ▁algebra ▁is ▁in ▁fact ▁unique ▁up ▁to ▁isomorphism . ▁Some ▁authors ▁use ▁" von ▁Ne umann ▁algebra " ▁for ▁the ▁al gebras ▁together ▁with ▁a ▁Hil bert ▁space ▁action , ▁and ▁" W * - algebra " ▁for ▁the ▁abstract ▁concept , ▁so ▁a ▁von ▁Ne umann ▁algebra ▁is ▁a ▁W * - algebra ▁together ▁with ▁a ▁Hil bert ▁space ▁and ▁a ▁suitable ▁faithful ▁un ital ▁action ▁on ▁the ▁Hil bert ▁space .