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,