SECTION 1: INTRODUCTION

Suppose that G is a compact Lie group and that A and B are

C -algebras upon which G acts (referred to here as G-algebras).

With modest hypotheses on A and B, G.G. Kasparov [Ka2] has

defined groups KK.(A,B), j € Z2, which are equivariant

generalizations of the renowned Kasparov groups KK^fA^) which

play a fundamental role in the modern theory of C -algebras. The

groups KK.(A,B) seem destined for a similar role. For example, if

n

M is a compact smooth manifold on which G acts by

diffeomorphisms, then the G-algebra of pseudo-differential

operators of order 0 on M determines an index element of

G KK^CfS*M)

,K) , where K = K(H) is the algebra of compact operators

on a Hilbert space with some G-action and S M is the cosphere

bundle. Determination of this element gives a strong form of the

Atiyah-Singer equivariant index theorem for families. Note that

the usual equivariant K-theory groups arise via the

identification

K^(B) 2 KKj(C,B),

and, in particular,

K~j(X) * KKj(C,C(X))

for X a compact G-space.

This paper is concerned, first of all, with effective

methods of computation for equivariant K-theory and KK-theory.

Received by the editors July 3, 1985.

Research partially supported by NSF grants 81-20790 (JR,CS), DMS

84-00900 (JR) and DMS 84-01367 (CS).

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