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