In this paper we investigate separation properties in the dual
Ĝ of a connected,
simply connected, nilpotent Lie group G. Following [4,
19], we are particularly
interested in the question of when the group G is quasi-standard,
in which case the group
C*-algebra C*(G) may be represented
as a continuous bundle of C*-algebras over
a locally compact, Hausdorff, space such that the fibres are primitive
throughout a
dense subset. The same question for other classes of locally compact groups
has been
considered previously in [1, 5, 18].
Fundamental to the study of quasi-standardness is the relation of inseparability
in
Ĝ[ratio ]π∼σ in Ĝ
if π and σ cannot be separated by disjoint open subsets of Ĝ.
Thus we have been led naturally to consider also the set
sep (Ĝ) of separated points in Ĝ (a point in a topological
space is separated if it can
be separated by disjoint open subsets from each point that is not in its
closure).