EQUIVARIANT E-THEORY FOR C*-ALGEBRAS 3
THEOREM.
If G is a countable discrete group and if the identity morphism
1 G EG{C, C) factors through a proper G-C*-algebra then, for any G-C*-algebra B,
the Baum-Connes assembly map
li:EG{£G,B)-K*(C*(G,B))
is an isomorphism.
The proof is very straightforward—if the identity on C factors through a proper
G-C*-algebra D then the assembly map for B identifies with a direct summand of
the assembly map for the proper G-C*-algebra B (g D, and by the generalized
Green-Julg theorem the latter is an isomorphism.
Throughout the preceding discussion we have used the full crossed product
C*-algebra C*(G,B) rather than its reduced counterpart C*ed(G, B). It is a defi-
nite shortcoming of ^-theory that it is not as well adapted to the reduced crossed
product as is Kasparov's if if-theory. On the other hand if a discrete group G is
C*-exact, in the sense that minimal tensor product with the reduced C*-algebra
C*ed(G) preserves short exact sequences of C*-algebras, then in the above discus-
sion we can replace the full with the reduced crossed product. Conjecturally all
discrete groups are C*-exact, and many classes of groups are known to be so (most
notably discrete subgroups of connected Lie groups), so from a practical perspec-
tive this shortcoming of £?-theory is perhaps not so great. To reinforce this point,
we note that in the key applications of the Baum-Connes theory to topology and
geometry (via, for instance, the Novikov conjecture) it is sufficient to work with full
crossed product C*-algebras [27]. But the incompatibility of ^-theory with reduced
crossed products is nonetheless an awkward circumstance. It suggests that the ma-
chinery developed in this paper will not be the final and most suitable framework
for the Baum-Connes theory—but this is a speculation which must be enlarged
upon elsewhere.
While this paper was in the final stages of preparation we received from Klaus
Thomsen an interesting paper [43] on essentially the same subject. There is less
overlap between the two articles than one might expect: as we have noted above,
the main emphasis of the present work is the applications of i£-theory to the Baum-
Connes conjecture, whereas Thomsen's article is concerned more with foundational
questions concerning homology-type functors on the category of G-C*-algebras (see
also [44] for a result relevant to the relationship between ^-theory and Kasparov's
if if-theory). In fact the papers complement one another quite nicely, although we
note that certain basic objects, such as the equivariant ^-theory groups themselves,
are defined differently in the two papers. There is also an interesting overlap be-
tween E'-theory and Cuntz's approach to bivariant if-theories [13]; this is explained
in [14]. But once again, our emphasis here is rather different.
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