EQUIVARIANT K-THEORY AND KK-THEORY n
"Borel cochain" cohomology theory) iff
H2(G,Z)
= 0 iff
3
H = 0 (where top denotes usual topological (singular) top(BG,Z)
^
cohomology). However, H (BG;Z) vanishes modulo torsion in odd
degrees, since if T is a maximal torus in G and W is the
corresponding Weyl group, then
H*(BG,D) S
H*(BT,D)W
and H (BT,G) is a polynomial algebra on generators in degree 2.
Therefore, by the classical universal coefficient theorem,
H3(BG,Z) 2 Tors H2(BG,Z)
where Tors denotes the torsion subgroup. Since G is connected, BG
is simply-connected and, by the Hurewicz theorem,
Tors H2(BG,Z) S Tors *2(BG) 2 Tors ^(G). a
DEFINITION 2.2. A collection of G-algebras F is a
CG~fundamental family if every Type I G-algebra may be
constructed from elements of F by taking extensions, kernels,
quotients, inductive limits, tensor product with the trivial
G-algebra C (IR), by changing the G-action up to exterior
equivalence, and G-stable isomorphism. (The operations may be
applied any countable number of times in any order.) Similarly, F
is called an AG-fundamental family if every abelian G-algebra may
be constructed out of F.
For instance, if the group G is trivial, then the family
{C (I R )QK) is a fundamental family, by the well-known structure
theory of Type I algebras (cf. [Sc2, 12]). In fact, we may be
even more frugal and use the one-element family {£}, since we are
allowed to tei
tensor with K,
allowe d t o tensor with C (IR ) , hence with C (lRn) , and since we may
We recall some basic information on the C -algebra
associated to a group and a cocycle. Suppose that G is a locally
2
compact abelian group, and suppose that w Z (G,I) is a
normalized cocycle. Then we form L (G,w) which is just L (G) as a
Banach space, but with a convolution product twisted by w. It is
the universal object for maps *:G U(H) with the property
that
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