Let G be a locally compact
group and G0 its connected component of the identity. If G∕G0 is compact, then G is
a projective limit of Lie groups. In fact, there exist arbitrarily small compact normal
subgroups H ⊆ G such that G∕H is a Lie group. Suppose H is such a compact,
co-Lie subgroup of G. Then any unitary representation of G∕H can be lifted to G
in a natural way. Conversely, given a unitary representation π of G, one
may ask whether it really lives on a Lie factor—that is, does there always
exist a compact normal subgroup H ⊆ G such that G∕H is a Lie group and
π(h),h ∈ H, is the identity operator ? In this paper it is shown that this is
indeed the case whenever π is irreducible (or more generally whenever π is
factorial). The dual space Ĝ (= equivalence classes of irreducible unitary
representations) is then realized as an inductive limit of the dual spaces of Lie
groups. This inductive limit is first cast in a topological setting (using the
dual topology on Ĝ); and then, when G is also unimodular and type I, one
obtains a measure-theoretic interpretation of the inductive limit (using the
Plancherel measure). One application of these results is the fact that an almost
connected group whose solvable radical is actually nilpotent must be a type I
group.
