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
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