AbstactA necessary and sufficient condition is given for a separable C*-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation.

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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).

Let A be a unital Banach function algebra with character space Φ A . For x ∈ Φ A , let M x and J x be the ideals of functions vanishing at x, and in a neighbourhood of x, respectively. It is shown that the hull of J x is connected, and that if x does not belong to the Shilov boundary of A then the set {y ∈ Φ A : M x ⊇ J y } has an infinite, connected subset. Various related results are given. Maths Reviews Classification 46J20