7r(s)*(t) = w(s,t)7c(s+t) .
Then C (G,w) is the usual completion. Its isomorphism class only
depends upon the cohomology class of a in H (G,T). Note that
C (G,u) is unital if and only if G is a discrete group.
The cocycle w is called totally skew if u(x,y) = w(y,x) for
all y G implies x = 0. Changing w within its cohomology class
if necessary, we may always assume w is lifted from a totally
skew multiplier on a quotient group G/K, where K is uniquely
determined by the cohomology class of w [BK, Theorem 3.1]. Then w
determines a continuous injection h : G/K (G/K) with dense
range, and w is Type I if and only if h is bicontinuous [BK,
Theorem 3.2]. We record here the following general result due to
Baggett and Kleppner [BK], Kleppner (unpublished), and Pukanszky
[Pu, Ch. I, Proposition 2.1]; see also Green [Grl, Prop. 33].
THEOREM 2.3. Let G be a locally compact abelian group and let u
be a totally skew cocycle on G. The C -algebra C (G,w) is simple,
and it is Type I if and only if it is isomorphic to K(H) for some
Hilbert space H. If so, and if G is a discrete abelian group then
K(H) is unital, hence H has finite dimension and G is finite.
Our interest lies in the Type I setting. So suppose that F
is a free abelian group of finite rank with cocycle u . Let K
denote the radical of the associated skew form. Then C (F/K,w) is
a unital algebra. If it is Type I then Theorem 2.3 implies that
F/K is a finite group, so that K is of finite index in F. We
record this as a corollary.
COROLLARY 2.4. Suppose that F is a free abelian group of finite
rank with a Type I cocycle w. Then the radical K of the
associated skew form has finite index in F, and F/K 2 CXC for
some finite group C with w the dual pairing CXC T. D
If G = Tn is a torus with nl and if H is a closed subgroup,
then it is not necessarily true that the group HW(G,C(G/H,T)), in
which obstructions to exterior equivalence of two G-actions on an
algebra with G-spectrum G/H live [RR, Corollary 0.13], must
vanish. (See [Ro3].) Nevertheless, we can prove the following:
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