EQUIVARIANT K-THEORY AND KK-THEORY

13

THEOREM 2.5. Let G be a torus (of any dimension) and let

a:G Aut(A)

be an action of G on a continuous-trace algebra A such that

A 2 G/H as a G-space for some closed subgroup H of G. Then the

Dixmier-Douady class of A is trivial, i.e., A is stably

isomorphic (as an algebra) to C(G/H)$lf. Further, the action a$id

is exterior equivalent to £#id, where B is the natural action of

*

G on C

(K1,a)

and KSH is an appropriate subgroup. Thus, up to

stable isomorphism and exterior equivalence, A is G-isomorphic to

C{G/H,w)®lf. Here C(G/H,u) denotes IndHtQEnd(V), where V is an

irreducible ^-representation of H.

PROOF: Since G acts transitively on A with isotropy group H, the

Mackey machine implies that

(AxaG)~ a (H,ur,

2

where u € H (H,T) is the Mackey obstruction. Regardless of H and

•»

w, (H,u) is a countable set. Thus, replacing (A,a) by (A®K,a$id)

if necessary, we have AxG isomorphic to a countable direct sum of

copies of K. Since we have stabilized A, Takai duality implies

that

A 2 (AxaG)xQG.

Here G is a free abelian group. By the Mackey machine again, A

may be computed in terms of the G-orbits on the countable set

(A^G)".

Since A 2 (G/H) is connected, there can be only one G-orbit,

for otherwise Ax G would split as a direct sum of two G-invariant

ideals, which would give a non-trivial decomposition of A as a

disjoint union of two connected components. Let K be the common

stability group in G of the points in (Ax G) . (By Pontrjagin

duality, each subgroup of G is the annihilator of a unique closed

2

subgroup of G.) Then if o r € H

(Kx,T)

is the Mackey obstruction

for the action of K on Ax G, we have