Let G be a locally compact
group, ω ∈ Z2(G, 𝕋) a (measurable) multiplier on G, and denote by C∗(G,ω) the
twisted group C∗-algebra of G defined by ω. We are only interested in multipliers up
to equivalence, so we always tacitly assume that one is free to vary a multiplier
within its cohomology class in H2(G, 𝕋). In this paper we are basically concerned
with the following two problems: the first is to determine the structure of C∗(G,ω),
where ω is a type I multiplier on the locally compact abelian group G, and the
second is to describe the crossed product A ⋊αG of a continuous-trace C∗-algebra A
by an action of an abelian group G, such that the corresponding action of G
on  has constant stabilizer N, and the Mackey obstruction to extending
an irreducible representation ρ of A to a covariant representation (ρ,U) of
(A,N,αN) is equal to a constant multiplier ω ∈ H2(N, 𝕋) for all ρ ∈Â. The
second of these problems is the obvious starting point for the study of the
“fine structure of the Mackey machine”, for actions of abelian groups on
continuous-trace algebras with “continuously varying” stabilizers and Mackey
obstructions.
