14
JONATHAN ROSENBERG AND CLAUDE SCHOCHET
A S (K1,*)".
Since A is type I, a is also Type I. Let L be the radical of the
*
associated form. Then C
(K1/L,a)
is Type I and unital. By
i
* »
Corollary 2.4, L has finite index in K . Thus (since A = G/H)
dim (G/H) = dim (K1)"= dim (G/K)
and dim H = dim K. In fact, as the G-action on A must be dual to
*»»
the action of G on (Ax G), we see that H = K if a is trivial and,
more generally, H = L. Since (Ax G) is discrete, each
spectrum-fixing automorphism of Ax G is inner [RR, Theorem
0.5(b)], and by [RR, Theorem 0.11] the only obstruction to
exterior equivalence of the action of G on Ax G with the standard
action of G on
C(G/KL,K)
is a class in
H2(G,C{(AxaG)~,I)).
Now
H2(G,C((AxaGT,T))
=
H2(G,C(G/K1,T))
=
H2(G,U(G/KX,T))
(in the notation of Moore [Mo], this being true since G/K is
discrete)
* H2(K1,T))
by [Mo, Theorem 6] (Moore's version of Shapiro's Lemma), and our
obstruction class corresponds to a. Thus A is stably isomorphic
*
to C
(Kl,a).
Now this is a continuous-trace algebra with spectrum
G/H and fibres isomorphic to End(V) , where V is the unique (by
the Stone-von Neumann theorem) irreducible ^-representation of
the finite group KVH 1 S (H/K)". Since End(V) is
*
finite-dimensional, the Dixmier-Douady class of C
(K1,o)
must be
3
a torsion class in H (G/H,Z) (by an observation of Serre [Gr3]).
However, G/H is a torus, so its cohomology is torsion-free. Thus
the Dixmier-Douady class of A vanishes.
In fact, we have proved somewhat more. By Takai duality, not
only is
AQK s s C (K1,^)^
but also the action ct$id must be exterior equivalent to B$id,
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