▁von ▁Ne umann ▁algebra , ▁indeed ▁a ▁factor , ▁of ▁type ▁I . ▁If ▁we ▁have ▁any ▁unit ary ▁representation ▁of ▁a ▁group ▁G ▁on ▁a ▁Hil bert ▁space ▁H ▁then ▁the ▁bounded ▁operators ▁comm uting ▁with ▁G ▁form ▁a ▁von ▁Ne umann ▁algebra ▁G ′ , ▁whose ▁pro jections ▁correspond ▁exactly ▁to ▁the ▁closed ▁sub spaces ▁of ▁H ▁invariant ▁under ▁G . ▁Equ ivalent ▁sub re present ations ▁correspond ▁to ▁equivalent ▁pro jections ▁in ▁G ′ . ▁The ▁double ▁commut ant ▁G ′ ′ ▁of ▁G ▁is ▁also ▁a ▁von ▁Ne umann ▁algebra . ▁ ▁The ▁von ▁Ne umann ▁group ▁algebra ▁of ▁a ▁discrete ▁group ▁G ▁is ▁the ▁algebra ▁of ▁all ▁bounded ▁operators ▁on ▁H ▁= ▁l 2 ( G ) ▁comm uting ▁with ▁the ▁action ▁of ▁G ▁on ▁H ▁through ▁right ▁multiplication . ▁ ▁One ▁can ▁show ▁that ▁this ▁is ▁the ▁von ▁Ne umann ▁algebra ▁generated ▁by ▁the ▁operators ▁corresponding ▁to ▁multiplication ▁from ▁the ▁left ▁with ▁an ▁element ▁g ▁ ∈ ▁G . ▁It ▁is ▁a ▁factor ▁( of ▁type ▁II 1 ) ▁if ▁every ▁non - t rivial ▁conjug acy ▁class ▁of ▁G ▁is ▁infinite ▁( for ▁example , ▁a ▁non - abel ian ▁free ▁group ), ▁and ▁is ▁the ▁hyper finite ▁factor ▁of ▁type ▁II 1 ▁if ▁in ▁addition ▁G ▁is ▁a ▁union ▁of ▁finite ▁sub groups ▁( for ▁example , ▁the ▁group ▁of ▁all ▁permut ations ▁of ▁the ▁integers ▁fixing ▁all ▁but ▁a ▁finite ▁number ▁of ▁elements ). ▁The ▁tensor ▁product ▁of ▁two ▁von ▁Ne umann ▁al gebras