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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