2 ERIK GUENTNER, NIGEL HIGSON, AND JODY TROU T

EXCISION.

If I \ A and J B are G-C*-ideals then the sequences

EG(A, J) - EG(A, B) - EG(A, B/J)

EG(LB) +- EG(A,B) - EG(A/I,B)

are exact in the middle.

STABILIZATION.

If p\A — B is a *-homomorphism of G-C*-algebras and if

for some G-Hilbert space Ji, the tensor product p 0 1: A ® %(%) - 5 0 %{%) is a

G-homotopy equivalence {where %(%) denotes the C*-algebra of compact operators

on %) then the morphism [p] G EG(A,B) is invertible.

It is not hard to see that EG is the universal theory with these properties. Of

course, a further key property is that an (equivariant) asymptotic morphism from

A to B determines an element of EG(A,B)—the whole point of EG-theory is to

provide a framework for calculations involving asymptotic morphisms. Finally, for

applications to C*-algebra If-theory the following property, which concerns the full

crossed product C*-algebra C*(G,A), is crucial:

DESCENT.

There is a functor from the equivariant E-theory category to the

non-equivariant E-theory category, mapping the class in EG(A1B) of an equivar-

iant *-homomorphism from A to B to the class in E(C*(G, A), C*(G, B)) associated

to the induced *-homomorphism from C*(G,A) to C*(G,B).

The later chapters of our paper borrow from the second author's collaboration

with Kasparov, to whom we are grateful for allowing us to include some of that

joint work here. Following roughly the procedure in [4] we define the Baum-Connes

assembly map

Lt:EG{£G,B)^K*(C*(G,B)).

If G is compact then by adapting to EG-th.eovy a well-known argument of Green [18]

and Julg [24] we prove that the assembly map is an isomorphism. In order to study

assembly for non-compact groups we follow Kasparov's lead [26] and introduce a

notion of proper G-C*-algebra. A guiding principle is that the action of a non-

compact group on a proper G-C*-algebra is roughly the same as a compact group

action, and with this in mind we seek to generalize the Green-Julg isomorphism to

proper G-C*-algebras. Unfortunately some technical obstacles arise, but we are at

least able to prove the following result:

GENERALIZED GREEN-JULG THEOREM.

IfG is a countable discrete group and

if D is a proper G-C*-algebra then the Baum-Connes assembly map

F.EG{£G,D) ^ K.(C*(G,D))

is an isomorphism.

Although we shall not go into it here, the theorem is also true for a variety of

other classes of groups (for example, connected Lie groups and totally disconnected

groups). However it is not clear to us that the statement is correct for general

locally compact groups, particularly for groups of infinite dimension.

In any case, concentrating on discrete groups, we obtain the following important

result which is central to the paper [22] and which might be regarded as the focus

of the present article: