▁bounded ▁operators ▁on ▁some ▁Hil bert ▁space ; ▁ ▁since ▁there ▁is ▁one ▁Hil bert ▁space ▁for ▁every ▁cardinal ▁number , ▁isomorphism ▁classes ▁of ▁factors ▁of ▁type ▁I ▁correspond ▁exactly ▁to ▁the ▁cardinal ▁numbers . ▁ ▁Since ▁many ▁authors ▁consider ▁von ▁Ne umann ▁al gebras ▁only ▁on ▁separ able ▁Hil bert ▁spaces , ▁it ▁is ▁custom ary ▁to ▁call ▁the ▁bounded ▁operators ▁on ▁a ▁Hil bert ▁space ▁of ▁finite ▁dimension ▁n ▁a ▁factor ▁of ▁type ▁In , ▁and ▁the ▁bounded ▁operators ▁on ▁a ▁separ able ▁infinite - dimensional ▁Hil bert ▁space , ▁a ▁factor ▁of ▁type ▁I ∞ . ▁ ▁Type ▁II ▁factors ▁A ▁factor ▁is ▁said ▁to ▁be ▁of ▁type ▁II ▁if ▁there ▁are ▁no ▁minimal ▁pro jections ▁but ▁there ▁are ▁non - zero ▁finite ▁pro jections . ▁This ▁implies ▁that ▁every ▁projection ▁E ▁can ▁be ▁“ hal ved ” ▁in ▁the ▁sense ▁that ▁there ▁are ▁two ▁pro jections ▁F ▁and ▁G ▁that ▁are ▁Murray – von ▁Ne umann ▁equivalent ▁and ▁satisfy ▁E ▁= ▁F ▁+ ▁G . ▁ ▁If ▁the ▁identity ▁operator ▁in ▁a ▁type ▁II ▁factor ▁is ▁finite , ▁the ▁factor ▁is ▁said ▁to ▁be ▁of ▁type ▁II 1 ; ▁otherwise , ▁it ▁is ▁said ▁to ▁be ▁of ▁type ▁II ∞ . ▁ ▁The ▁best ▁understood ▁factors ▁of ▁type ▁II ▁are ▁the ▁hyper finite ▁type ▁II 1 ▁factor ▁and ▁the ▁hyper finite ▁type ▁II ∞ ▁factor , ▁found ▁by ▁. ▁These ▁are ▁the ▁unique ▁hyper finite ▁factors ▁of ▁types ▁II 1 ▁and ▁ ▁II ∞ ; ▁there ▁are ▁an ▁un