▁trivial ▁representation ▁is ▁isolated ▁in ▁the ▁dual ▁space ), ▁such ▁as ▁SL ( 3 , Z ), ▁ ▁has ▁a ▁count able ▁fundamental ▁group . ▁Sub sequently , ▁Sor in ▁Pop a ▁showed ▁that ▁the ▁fundamental ▁group ▁can ▁be ▁trivial ▁for ▁certain ▁groups , ▁including ▁the ▁sem id irect ▁product ▁of ▁Z 2 ▁by ▁SL ( 2 , Z ). ▁ ▁An ▁example ▁of ▁a ▁type ▁II 1 ▁factor ▁is ▁the ▁von ▁Ne umann ▁group ▁algebra ▁of ▁a ▁count able ▁infinite ▁discrete ▁group ▁such ▁that ▁every ▁non - t rivial ▁conjug acy ▁class ▁is ▁infinite . ▁ ▁found ▁an ▁un count able ▁family ▁of ▁such ▁groups ▁with ▁non - is omorphic ▁von ▁Ne umann ▁group ▁al gebras , ▁thus ▁showing ▁the ▁existence ▁of ▁un count ably ▁many ▁different ▁separ able ▁type ▁II 1 ▁factors . ▁ ▁Type ▁III ▁factors ▁Last ly , ▁type ▁III ▁factors ▁are ▁factors ▁that ▁do ▁not ▁contain ▁any ▁non zero ▁finite ▁pro jections ▁at ▁all . ▁ ▁In ▁their ▁first ▁paper ▁ ▁were ▁unable ▁to ▁decide ▁whether ▁or ▁not ▁they ▁existed ; ▁the ▁first ▁examples ▁were ▁later ▁found ▁by ▁. ▁Since ▁the ▁identity ▁operator ▁is ▁always ▁infinite ▁in ▁those ▁factors , ▁they ▁were ▁sometimes ▁called ▁type ▁III ∞ ▁in ▁the ▁past , ▁but ▁recently ▁that ▁notation ▁has ▁been ▁su pers eded ▁by ▁the ▁notation ▁III λ , ▁where ▁ λ ▁is ▁a ▁real ▁number ▁in ▁the ▁interval ▁[ 0 , 1 ]. ▁More ▁precisely , ▁if ▁the ▁Con nes ▁spectrum ▁( of ▁its ▁mod ular ▁group ) ▁is ▁