Introduction

The notion of an asymptotic morphism between two C*-algebras was introduced

in a brief note of Connes and Higson [11]. An asymptotic morphism from A to B

induces a homomorphism from the C*-algebra if-theory of A to that of B. At the

level of homotopy there is a composition law for asymptotic morphisms which is

compatible with if-theory. The homotopy category of asymptotic morphisms so

obtained is a powerful tool, closely related to Gennadi Kasparov's if if-theory [25],

for calculating C*-algebra if-theory groups.

The purpose of this

article1

is to develop in some detail the theory of equivari-

ant asymptotic morphisms, appropriate to C*-algebras equipped with continuous

actions of locally compact groups, and so construct tools very similar to those

of Kasparov's equivariant if if-theory [26] for calculating the if-theory of group

C*-algebras. A central problem in C*-algebra if-theory is the Baum-Connes con-

jecture [4], which proposes a formula for the if-theory of group C*-algebras. A

primary goal of the paper is to first formulate the conjecture in the language of

asymptotic morphisms, and then describe a general method, due essentially to

Kasparov, for attacking various cases of it. At present the method encompasses

nearly all that is known about the Baum-Connes conjecture. The method can be

implemented within either our theory or Kasparov's, but we note that the most

recent progress on the conjecture [22] does not yet fit fully into the framework of

equivariant if if-theory, so for the time being the theory of asymptotic morphisms

appears to be an essential variation on Kasparov's work. We shall not describe

how our approach to the Baum-Connes conjecture applies to the computation of

if-theory for specific group C*-algebras—for this we refer the reader to the pa-

per [22]. Instead we concentrate on providing a reasonably conceptual framework

for that and other investigations into the if-theory of group C*-algebras.

Our starting point is a definition of asymptotic morphism which differs slightly

from the one introduced by Connes and Higson [11] and is presented in Chapter 1.

Chapters 2-5 develop properties of the homotopy category of asymptotic morphisms

and in Chapter 6 we introduce the equivariant ^-theory groups

EQ{A,B).

In

Chapter 7 we summarize the homological properties of jE'c-theory. In brief, these

are as follows:

COMPOSITION PRODUCT.

The abelian groups

EQ(A,

B) are the morphism sets

in an additive category whose objects are the separable G-C* -algebras. There is a

functor into this category from the homotopy category of G-C*-algebras and equi-

variant *-homomorphisms.

1 Received by the editor November 17, 1997.

1