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).

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Following the work of Allen, Sinclair and Smith on the primitive ideal space of the Haagerup tensor product A ⊗ hB of C*-algebras A and B, we investigate the hull-kernel topology and use this to determine various other ideal spaces and their topologies in relation to the corresponding ideal spaces of A and B. We study the semi-continuity of norm functions I → ||x + I||(x ∈ A ⊗h B) on these ideal spaces and identify the separated points of Prim (A ⊗h B). Finally, we exhibit several conditions each of which is equivalent to the quasi-standardness of A ⊗h B.