10 JONATHAN ROSENBERG AND CLAUDE SCHOCHET

G-extensions, inductive limits, crossed products by IR - or

Z-actions commuting with the G-action, exterior equivalence, and

G-stable isomorphism. BQ denotes the "bootstrap category", the

smallest subcategory of CQ which contains AQ and is closed under

the same operations as Cr. Since we are dealing with nuclear

C -algebras, all tensor products may be taken to be spatial. Then

if A and B are G-algebras, G acts on A#B by the diagonal action.

A and B are said to be G-stably isomorphic if A$K s B$K, where s

means "G-equivariant *-isomorphism". By Kasparov's stabilization

theorem [Kal] (see also (3.1) below), the question of which

unitary representation of G is used in constructing the G-action

on K is irrelevant, and we may take the action to be an infinite

2

number of copies of the regular representation on L (G).

We say that the group G satisfies the Hodgkin condition if G

is connected and if ^-(G) is torsion-free. As pointed out by

Hodgkin [Ho, p. 68], this condition implies but is strictly

stronger than assuming that R(G) has finite global homological

dimension. The global dimension of R(G) is then rank(G)+l.

Groups satisfying the Hodgkin condition have another very

important property. If T is a maximal torus of G, then R(T) is a

free R(G)-module of finite rank (by [Pi] and [St]), and hence the

ring extension R(G) * • R(T) is faithfully flat. We shall use this

fact (cf. Theorem 3.7) to reduce many questions about K-theory of

G-algebras to questions about T-algebras. Accordingly, it is

useful to introduce the category BQ of G-algebras in CQ which,

when viewed as T-algebras, lie in B_. Note that

AG C BG C 5G S CG-

The Hodgkin condition on G might seem a little mysterious.

However, from our point of view, it has a very natural

explanation given in the following proposition.

PROPOSITION 2.1. Let G be a compact connected Lie group. Then G

has no non-trivial projective representations if and only if

Kj(G) is torsion-free.

PROOF: By [Mo] and [Wi], G has no non-trivial projective

2

representations iff HM(G,T) = 0, (where M indicates* Moore's