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
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