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.
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:
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
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:
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:
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