▁two ▁seem ingly ▁different ▁points ▁of ▁view . ▁ ▁B im od ules ▁are ▁also ▁important ▁for ▁the ▁von ▁Ne umann ▁group ▁algebra ▁M ▁of ▁a ▁discrete ▁group ▁ Γ . ▁Indeed , ▁if ▁V ▁is ▁any ▁unit ary ▁representation ▁of ▁ Γ , ▁then , ▁regarding ▁ Γ ▁as ▁the ▁diagonal ▁subgroup ▁of ▁ Γ ▁× ▁ Γ , ▁the ▁corresponding ▁induced ▁representation ▁on ▁l 2 ▁( Γ , ▁V ) ▁is ▁naturally ▁a ▁b im od ule ▁for ▁two ▁comm uting ▁copies ▁of ▁M . ▁Import ant ▁representation ▁the or etic ▁properties ▁of ▁ Γ ▁can ▁be ▁form ulated ▁entirely ▁in ▁terms ▁of ▁b im od ules ▁and ▁therefore ▁make ▁sense ▁for ▁the ▁von ▁Ne umann ▁algebra ▁itself . ▁For ▁example , ▁Con nes ▁and ▁Jones ▁gave ▁a ▁definition ▁of ▁an ▁analog ue ▁of ▁Kaz hd an ' s ▁property ▁( T ) ▁for ▁von ▁Ne umann ▁al gebras ▁in ▁this ▁way . ▁ ▁Non - amen able ▁factors ▁Von ▁Ne umann ▁al gebras ▁of ▁type ▁I ▁are ▁always ▁am enable , ▁but ▁for ▁the ▁other ▁types ▁there ▁are ▁an ▁un count able ▁number ▁of ▁different ▁non - amen able ▁factors , ▁which ▁seem ▁very ▁hard ▁to ▁class ify , ▁or ▁even ▁distinguish ▁from ▁each ▁other . ▁Nevertheless , ▁Vo ic ules cu ▁has ▁shown ▁that ▁the ▁class ▁of ▁non - amen able ▁factors ▁coming ▁from ▁the ▁group - measure ▁space ▁construction ▁is ▁dis joint ▁from ▁the ▁class ▁coming ▁from ▁group ▁von ▁Ne umann ▁al gebras ▁of ▁free ▁groups . ▁Later