DIMENSIONS AN D C*-ALGEBRA S 5 The geometrical an d analytica l dimensio n theorie s naturall y interac t i n th e Atiyah - Singer Inde x Theore m and its generalizations t o noncompact manifold s (see [13 ] an d Connes ' index theorem s fo r foliation s [15]) . I n these theorems , one relate s the analyti c inde x (de- fined i n terms o f Hilber t spac e dimensions ) o f a suitable pseudodifferentia l operato r o n a manifold t o a topological inde x o f th e symbo l o f th e operato r an d th e geometr y o f th e man - ifold. A puzzling aspec t o f th e abov e theorie s i s that i n many situation s th e rea l number s ar e either to o larg e o r inappropriat e fo r th e value s of dimension s and indices . Th e forme r possi - bility i s suggested b y Bohr' s observatio n regardin g th e windin g number s o f almos t periodi c functions. Th e algebraist s have foun d tha t K theor y provide s the natura l range s fo r mor e general dimensio n functions . Specifically , give n a ring R, th e suggeste d rang e i s the grou p KQ(R). Fo r inde x theor y on e ma y tak e R t o b e a n appropriat e C*~algebra, th e latte r bein g the continuous generalization of a semisimpie algebra, an d thu s geometrically mor e suitabl e than th e correspondin g vo n Neuman n algebras . As we have indicated , thes e lecture s ar e devote d t o th e K theor y o f C*-algebras , a subject whic h i s currently unde r intensiv e investigation . W e shall be largely concerne d wit h the ver y simples t o f C*-algebras , the approximately finite (AF ) C*-algebras o f Brattel i [8] . These are th e C*-algebrai c direc t limit s of finit e dimensiona l C*-algebras , th e latte r bein g just th e classica l semisimpi e algebras . I n contrast t o th e commutativ e context , wher e suc h limit algebra s are essentiall y trivial , the A F algebras ar e of considerabl e interest . Asid e fro m the "UHF" an d "matroi d algebras" , initially studie d b y Glim m [38 ] an d Dixmie r [24] , the AF algebras have resisted classification . C*-algebrai c dimensio n theor y wa s first explicitl y studied b y Dixmie r [24] , who use d i t t o classif y th e matroi d algebras . Elliot t the n extende d this theory t o th e A F algebras. I n the K theoreti c terminolog y (introduce d i n [28] , [41]) , he prove d th e strikin g resul t tha t th e K Q grou p o f a n A F algebra ha s a natural ordering , and that th e correspondin g ordere d grou p togethe r wit h a "scale" (see §7 ) provide s a complete algebraic invarian t fo r th e A F algebras. Th e ordere d group s tha t aris e ar e calle d dimension groups. I n many case s the latte r hav e prove d t o b e muc h easie r t o wor k wit h tha n th e A F algebras. I n particula r thei r structur e ha s interesting relation s t o algebrai c numbe r theory . As we indicated i n th e introductio n muc h o f th e theor y fo r A F algebras i s equally valid fo r th e locally semisimpie algebras. Furthermore , i n this contex t on e ma y tak e a much more elementar y approac h t o A' theory, defining th e dimensio n grou p K0 t o b e a suitable direct limit . Thu s Chapter s 2- 7 ar e almos t completel y devote d t o thes e algebras , C*-alge- braic K theor y appearin g firs t i n Chapte r 8 .
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