▁of ▁the ▁form ▁a * a ) ▁to ▁[ 0 , ∞ ]. ▁A ▁positive ▁linear ▁functional ▁is ▁a ▁weight ▁with ▁ ω ( 1 ) ▁finite ▁( or ▁rather ▁the ▁extension ▁of ▁ ω ▁to ▁the ▁whole ▁algebra ▁by ▁linear ity ). ▁A ▁state ▁is ▁a ▁weight ▁with ▁ ω ( 1 )   =   1 . ▁A ▁trace ▁is ▁a ▁weight ▁with ▁ ω ( aa *)   =   ω ( a * a ) ▁for ▁all ▁a . ▁A ▁tra cial ▁state ▁is ▁a ▁trace ▁with ▁ ω ( 1 )   =   1 . ▁ ▁Any ▁factor ▁has ▁a ▁trace ▁such ▁that ▁the ▁trace ▁of ▁a ▁non - zero ▁projection ▁is ▁non - zero ▁and ▁the ▁trace ▁of ▁a ▁projection ▁is ▁infinite ▁if ▁and ▁only ▁if ▁the ▁projection ▁is ▁infinite . ▁Such ▁a ▁trace ▁is ▁unique ▁up ▁to ▁res cal ing . ▁ ▁For ▁factors ▁that ▁are ▁separ able ▁or ▁finite , ▁two ▁pro jections ▁are ▁equivalent ▁if ▁and ▁only ▁if ▁they ▁have ▁the ▁same ▁trace . ▁The ▁type ▁of ▁a ▁factor ▁can ▁be ▁read ▁off ▁from ▁the ▁possible ▁values ▁of ▁this ▁trace ▁as ▁follows : ▁Type ▁In : ▁ 0 , ▁x , ▁ 2 x , ▁.... , nx ▁for ▁some ▁positive ▁x ▁( us ually ▁normal ized ▁to ▁be ▁ 1 / n ▁or ▁ 1 ). ▁Type ▁I ∞ : ▁ 0 , ▁x , ▁ 2 x , ▁.... , ∞ ▁for ▁some ▁positive ▁x ▁( us ually ▁normal ized ▁to