Introduction With the appearanc e o f Connes ' pape r o n C*-algebrai c generalization s o f th e Atiyah - Singer Index Theorem [17] , [70], "noncommutative" or "C*-algebraic" differential geometr y has become a reality. Thi s has been associated with the developmen t o f a branch o f noncommu- tative algebrai c topology, namely C*-algebrai c K 0 an d K t theory . Roughl y speaking , the former ma y be thought o f a s dimension theor y fo r projection s i n a C*-algebra an d it s matri x rings. I n these notes we have attempted t o giv e a reasonably self-containe d introductio n t o C*-algebraic K 0 theor y whic h will hopefull y b e accessibl e t o nonspecialists . Indee d th e reader doe s not nee d t o kno w wha t a C*-algebra i s until h e reache s Chapte r 8 . Du e to a lack o f time , we could no t discus s trace theory , the elegan t wor k o f Fac k an d Mardcha l [35] , the detaile d theor y o f th e Kx groups , cross-products, global idea l theory, o r the Ex t groups . We begin in Chapte r 1 with a heuristic explanatio n o f ho w C*-algebrai c dimension s naturally aris e in analysi s and geometry . Th e followin g si x chapters ar e essentially concerne d with dimension s in AF algebras. A s pointed ou t b y Elliot t [33 ] thi s theory ma y be place d in a purely algebrai c settin g by usin g algebraic direc t limit s o f finit e dimensiona l algebra s rather tha n th e correspondin g C*-algebrai c limit s (the A F algebras) . I n Chapte r 6 we give a brief introductio n t o Krieger's theor y o f dimension s i n topologica l dynamics . Chapter s 8 and 9 are devote d t o C*-algebrai c K theory , an d especiall y t o tha t fo r th e A F algebras . Finall y in Chapter 1 0 we briefly discus s some recen t result s an d ope n problems , including th e K theory o f cross-products . 1
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