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