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Dual topology of diamond groups. (English) Zbl 0833.22011

It is known that the Kirillov mapping \(\mathcal P\), which assigns to every primitive ideal \(J\) of the \(C^*\)-algebra of an exponential group \(G = \text{exp} ({\mathfrak g})\) its Kirillov orbit \({\mathcal P}(J) = \Omega \subset {\mathfrak g}^*\), is continuous. In this paper let \(\mathfrak g\) be a diamond Lie algebra, i.e. \(\mathfrak g\) is the semi-direct product \({\mathfrak g} = {\mathfrak a} \ltimes {\mathfrak h}\) of an abelian algebra \(\mathfrak a\) with a Heisenberg algebra \(\mathfrak h\), such that the spectrum of \(\text{ad} (X)\) is purely imaginary for every \(X \in {\mathfrak a}\) and such that the center of \(\mathfrak g\) is the center of \(\mathfrak h\), \(\mathfrak g\) is solvable but not exponential. We call the connected simply connected Lie group \(G\) associated with \(\mathfrak g\) a diamond group. In this paper we prove that the mapping \(\mathcal P\), which assigns to every primitive ideal \(J\) of the \(C^*\)-algebra of \(G\) its Kirillov-Pukanszky quasi orbit \({\mathcal P}(J) = \Omega \subset {\mathfrak g}^*\), is continuous. The main tool of the proof is a certain \(G\)-invariant polynomial on \({\mathfrak g}^*\), which appears in the description of the \(\text{Ad}^* (G)\)-orbits in general position and its associated central element in the enveloping algebra of \(\mathfrak g\).
Reviewer: J.Ludwig (Metz)

MSC:

22E25 Nilpotent and solvable Lie groups
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