count able ▁number ▁of ▁other ▁ ▁factors ▁of ▁these ▁types ▁that ▁are ▁the ▁subject ▁of ▁int ensive ▁study . ▁ ▁proved ▁the ▁fundamental ▁result ▁that ▁a ▁factor ▁of ▁type ▁II 1 ▁has ▁a ▁unique ▁finite ▁tra cial ▁state , ▁ ▁and ▁the ▁set ▁of ▁traces ▁of ▁pro jections ▁is ▁[ 0 , 1 ]. ▁ ▁A ▁factor ▁of ▁type ▁II ∞ ▁has ▁a ▁semif in ite ▁trace , ▁unique ▁up ▁to ▁res cal ing , ▁and ▁the ▁set ▁of ▁traces ▁of ▁pro jections ▁is ▁[ 0 , ∞ ]. ▁The ▁set ▁of ▁real ▁numbers ▁ λ ▁such ▁that ▁there ▁is ▁an ▁autom orphism ▁res cal ing ▁the ▁trace ▁by ▁a ▁factor ▁of ▁ λ ▁is ▁called ▁the ▁fundamental ▁group ▁of ▁the ▁type ▁II ∞ ▁factor . ▁ ▁The ▁tensor ▁product ▁of ▁a ▁factor ▁of ▁type ▁II 1 ▁and ▁an ▁infinite ▁type ▁I ▁factor ▁has ▁type ▁II ∞ , ▁and ▁convers ely ▁any ▁factor ▁of ▁ ▁type ▁II ∞ ▁can ▁be ▁constructed ▁like ▁this . ▁The ▁fundamental ▁group ▁of ▁a ▁type ▁II 1 ▁ ▁factor ▁is ▁defined ▁to ▁be ▁the ▁fundamental ▁group ▁of ▁its ▁tensor ▁product ▁with ▁the ▁infinite ▁( separ able ) ▁factor ▁of ▁type ▁I . ▁For ▁many ▁years ▁it ▁was ▁an ▁open ▁problem ▁to ▁find ▁a ▁type ▁II ▁factor ▁whose ▁fundamental ▁group ▁was ▁not ▁the ▁group ▁of ▁positive ▁re als , ▁but ▁Con nes ▁then ▁showed ▁that ▁the ▁von ▁Ne umann ▁group ▁algebra ▁of ▁a ▁count able ▁discrete ▁group ▁with ▁Kaz hd an ' s ▁property ▁( T ) ▁( the