▁be ▁▁ 1 ). ▁Type ▁II 1 : ▁[ 0 , x ] ▁for ▁some ▁positive ▁x ▁( us ually ▁normal ized ▁to ▁be ▁▁ 1 ). ▁Type ▁II ∞ : ▁[ 0 , ∞ ]. ▁Type ▁III : ▁{ 0 , ∞ }. ▁ ▁If ▁a ▁von ▁Ne umann ▁algebra ▁acts ▁on ▁a ▁Hil bert ▁space ▁containing ▁a ▁norm ▁ 1 ▁vector ▁v , ▁then ▁the ▁functional ▁a ▁→ ▁( av , v ) ▁is ▁ ▁a ▁normal ▁state . ▁This ▁construction ▁can ▁be ▁revers ed ▁to ▁give ▁an ▁action ▁on ▁a ▁Hil bert ▁space ▁from ▁a ▁normal ▁state : ▁this ▁is ▁the ▁G NS ▁construction ▁for ▁normal ▁states . ▁ ▁Mod ules ▁over ▁a ▁factor ▁Given ▁an ▁abstract ▁separ able ▁factor , ▁one ▁can ▁ask ▁for ▁a ▁classification ▁of ▁its ▁modules , ▁meaning ▁the ▁separ able ▁Hil bert ▁spaces ▁that ▁it ▁acts ▁on . ▁The ▁answer ▁is ▁given ▁as ▁follows : ▁every ▁such ▁module ▁H ▁can ▁be ▁given ▁an ▁M - dim ension ▁dim M ( H ) ▁( not ▁its ▁dimension ▁as ▁a ▁complex ▁vector ▁space ) ▁such ▁that ▁modules ▁are ▁is omorphic ▁if ▁and ▁only ▁if ▁they ▁have ▁the ▁same ▁M - dim ension . ▁The ▁M - dim ension ▁is ▁add itive , ▁and ▁a ▁module ▁is ▁is omorphic ▁to ▁a ▁sub space ▁of ▁another ▁module ▁if ▁and ▁only ▁if ▁it ▁has ▁smaller ▁or ▁equal ▁M - dim ension . ▁ ▁A ▁module ▁is ▁called ▁standard ▁if ▁it ▁has ▁a ▁cy clic ▁separ ating ▁vector . ▁Each ▁factor