▁Qu ot ien ing ▁out ▁deg ener acy ▁and ▁taking ▁the ▁completion ▁gives ▁a ▁Hil bert ▁space ▁▁▁▁ ▁whose ▁typical ▁element ▁is ▁an ▁equivalence ▁class ▁[ h ]. ▁For ▁a ▁fixed ▁g ▁in ▁G , ▁the ▁" shift ▁operator " ▁U g ▁defined ▁by ▁( U g )( h ) ▁( g ') ▁= ▁h ( g ▁− ▁g ), ▁for ▁a ▁representative ▁of ▁[ h ], ▁is ▁unit ary . ▁So ▁the ▁map ▁▁▁▁ ▁is ▁a ▁unit ary ▁representations ▁of ▁G ▁on ▁. ▁By ▁continu ity ▁of ▁f , ▁it ▁is ▁weak ly ▁continuous , ▁therefore ▁strongly ▁continuous . ▁By ▁construction , ▁we ▁have ▁▁▁▁▁ ▁where ▁[ e ] ▁is ▁the ▁class ▁of ▁the ▁function ▁that ▁is ▁ 1 ▁on ▁the ▁identity ▁of ▁G ▁and ▁zero ▁elsewhere . ▁But ▁by ▁G elf and – F ou rier ▁isomorphism , ▁the ▁vector ▁state ▁ ▁on ▁C *( G ) ▁is ▁the ▁pull - back ▁of ▁a ▁state ▁on ▁, ▁which ▁is ▁necessarily ▁integration ▁against ▁a ▁probability ▁measure ▁ μ . ▁Ch asing ▁through ▁the ▁is omorph isms ▁then ▁gives ▁▁▁▁ ▁On ▁the ▁other ▁hand , ▁given ▁a ▁probability ▁measure ▁ μ ▁on ▁, ▁the ▁function ▁▁▁▁ ▁is ▁a ▁normal ized ▁continuous ▁positive - defin ite ▁function . ▁Cont inu ity ▁of ▁f ▁follows ▁from ▁the ▁domin ated ▁convergence ▁theorem . ▁For ▁positive - def init eness , ▁take ▁a ▁non de generate ▁representation ▁of ▁. ▁This ▁extends ▁uniqu ely ▁to ▁a ▁representation ▁of ▁its ▁multip lier ▁algebra ▁ ▁and ▁therefore ▁a ▁strongly ▁continuous