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 10 we briefly discus s some recen t result s an d ope n problems , including th e K

theory o f cross-products .

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http://dx.doi.org/10.1090/cbms/046/01