▁Nar ut aka ▁O za wa ▁proved ▁that ▁group ▁von ▁Ne umann ▁al gebras ▁of ▁hyper bol ic ▁groups ▁yield ▁prime ▁type ▁II 1 ▁factors , ▁i . e . ▁ones ▁that ▁cannot ▁be ▁fact ored ▁as ▁tensor ▁products ▁of ▁type ▁II 1 ▁factors , ▁a ▁result ▁first ▁proved ▁by ▁Le em ing ▁Ge ▁for ▁free ▁group ▁factors ▁using ▁Vo ic ules cu ' s ▁free ▁entropy . ▁Pop a ' s ▁work ▁on ▁fundamental ▁groups ▁of ▁non - amen able ▁factors ▁represents ▁another ▁significant ▁advance . ▁The ▁theory ▁of ▁factors ▁" be yond ▁the ▁hyper finite " ▁is ▁rapidly ▁expand ing ▁at ▁present , ▁with ▁many ▁new ▁and ▁surprising ▁results ; ▁it ▁has ▁close ▁links ▁with ▁rig id ity ▁phen omena ▁in ▁geometric ▁group ▁theory ▁and ▁er god ic ▁theory . ▁ ▁Ex amples ▁The ▁essentially ▁bounded ▁functions ▁on ▁a ▁ σ - finite ▁measure ▁space ▁form ▁a ▁commut ative ▁( type ▁I 1 ) ▁von ▁Ne umann ▁algebra ▁acting ▁on ▁the ▁L 2 ▁functions . ▁For ▁certain ▁non - σ - finite ▁measure ▁spaces , ▁usually ▁considered ▁path ological , ▁L ∞ ( X ) ▁is ▁not ▁a ▁von ▁Ne umann ▁algebra ; ▁ ▁for ▁example , ▁the ▁ σ - algebra ▁of ▁meas urable ▁sets ▁might ▁be ▁the ▁count able - c oc ount able ▁algebra ▁on ▁an ▁un count able ▁set . ▁A ▁fundamental ▁approximation ▁theorem ▁can ▁be ▁represented ▁by ▁the ▁Kap l ans ky ▁density ▁theorem . ▁ ▁The ▁bounded ▁operators ▁on ▁any ▁Hil bert ▁space ▁form ▁a