ABSTRACT.

Let A and B be C*-algebras which are equipped with continuous ac-

tions of a second countable, locally compact group G. We define a notion of equi-

variant asymptotic morphism, and use it to define equivariant i£-theory groups

EQ(A, B) which generalize the ^-theory groups of Connes and Higson. We develop

the basic properties of equivariant E'-theory, including a composition product and

six-term exact sequences in both variables, and apply our theory to the problem

of calculating X-theory for group C*-algebras. Our main theorem gives a simple

criterion for the assembly map of Baum and Connes to be an isomorphism. The

result plays an important role in recent work of Higson and Kasparov on the Baum-

Connes conjecture for groups which act isometrically and metrically properly on

Hilbert space.

Key words and phrases, asymptotic morphisms,

Baum-Connes Conjecture, C*-algebras, equivariant ^-theory.

The first author was supported in part by NSF Grant DMS#97-06960; the

second author by NSF Grant DMS#95-00977; and the third author by NSF Grant

DMS#97-06767. This research was partially conducted during the period Nigel

Higson was employed by the Clay Mathematics Institute as a CMI Prize Fellow.