▁a ▁unique ▁probability ▁measure ▁ μ ▁on ▁. ▁Con vers ely , ▁the ▁Fourier ▁transform ▁of ▁a ▁probability ▁measure ▁on ▁ ▁is ▁necessarily ▁a ▁normal ized ▁continuous ▁positive - defin ite ▁function ▁f ▁on ▁G . ▁This ▁is ▁in ▁fact ▁a ▁one - to - one ▁correspond ence . ▁ ▁The ▁G elf and – F ou rier ▁transform ▁is ▁an ▁isomorphism ▁between ▁the ▁group ▁C * - algebra ▁C *( G ) ▁and ▁C 0 ( G ̂ ). ▁The ▁theorem ▁is ▁essentially ▁the ▁dual ▁statement ▁for ▁states ▁of ▁the ▁two ▁ab elian ▁C * - al gebras . ▁ ▁The ▁proof ▁of ▁the ▁theorem ▁passes ▁through ▁vector ▁states ▁on ▁strongly ▁continuous ▁unit ary ▁representations ▁of ▁G ▁( the ▁proof ▁in ▁fact ▁shows ▁that ▁every ▁normal ized ▁continuous ▁positive - defin ite ▁function ▁must ▁be ▁of ▁this ▁form ). ▁▁ ▁Given ▁a ▁normal ized ▁continuous ▁positive - defin ite ▁function ▁f ▁on ▁G , ▁one ▁can ▁construct ▁a ▁strongly ▁continuous ▁unit ary ▁representation ▁of ▁G ▁in ▁a ▁natural ▁way : ▁Let ▁F 0 ( G ) ▁be ▁the ▁family ▁of ▁complex - valu ed ▁functions ▁on ▁G ▁with ▁finite ▁support , ▁i . e . ▁h ( g ) ▁= ▁ 0 ▁for ▁all ▁but ▁finit ely ▁many ▁g . ▁ ▁The ▁positive - defin ite ▁kernel ▁K ( g 1 , ▁g 2 ) ▁= ▁f ( g 1 ▁− ▁g 2 ) ▁indu ces ▁a ▁( poss ibly ▁deg ener ate ) ▁inner ▁product ▁on ▁F 0 ( G ).