The primitive ideal space of two-step nilpotent group \(C^*\)-algebras (Q1340818)

For non-type I \(C^*\)-algebras \(A\) the general understanding is that it is impossible to determine the dual space \(\widehat {A}\), the set of equivalence classes of irreducible \(*\)-representations of \(A\). However, the primitive ideal space of \(A\), \(\text{Prim}(A)\), may well be a tractable dual object, as a set and topologically. Many people, including Baggett, Carey and Moran, Howe, Ludwig, Poguntke and the reviewer, have studied this problem for various classes of group \(C^*\)-algebras. In this very interesting paper the authors investigate the Jacobson topology on the primitive ideal space, \(\text{Prim}(C^*(N))\), of a 2- step nilpotent, second countable, locally compact group \(N\). A setwise description of \(\text{Prim}(C^*(N))\) has been known for some time. Indeed, \(\text{Prim}(C^*(N))\) can be parametrized by triples \((\lambda,N_ \lambda, \varphi)\), where \(\lambda \in \widehat{Z}\), the dual group of the centre \(Z\) of \(G\), \(N_ \lambda\) is the pullback of the centre of \(G/\text{ker }\lambda\) and \(\varphi\) runs through those characters of \(N_ \lambda\) which restrict to \(\lambda\) on \(Z\). \(\text{Prim}(C^*(N))\) is then given by the set of all kernels of induced representations \(\text{ker}(\text{ind}^ N_{N_ \lambda} \varphi)\). Using this parametrization the authors fully succeed in describing the topology on \(\text{Prim}(C^*(N))\) in terms of convergence of so-called subgroup representation pairs in the sense of Fell (Theorem 1.6). Certainly, a more global description of \(\text{Prim}(C^*(N))\) as a quotient space of some principal fibre bundle would be desirable, and the main concern of the paper is to find, under additional assumptions on \(N\), such a description. \(C^*(N)\) is known to decompose as the section algebra of a \(C^*\)- bundle over \(\widehat{Z}\), whose fibres are twisted group algebras for the abelian group \(A = N/Z\). Choose a Borel cross section \(\eta : A \to N\). Then the fibre over \(\lambda \in \widehat{Z}\) equals \(C^*(A,\sigma_ \lambda)\) where \(\sigma_ \lambda\) denotes the cocycle on \(A\) canonically associated to \(\lambda\) and \(\eta\). Now, the primitive ideal space of \(C^*(A,\sigma_ \lambda)\) is homeomorphic to \(\widehat{S}_ \lambda\), the dual group of the symmetrizer subgroup \(S_ \lambda\) of \(\sigma_ \lambda\). Thus there is a continuous and open mapping from \(\text{Prim}(C^*(N))\) onto \(\widehat{Z}\) whose fibre at \(\lambda \in \widehat{Z}\) is homeomorphic to \(\widehat{S}_ \lambda\). This gives a rough indication of how the problem of finding the desired fibre bundle may be approached. However, exploiting these ideas turns out to be fairly intricate and involves using various difficult results. The main result of the paper (Theorem 2.3) is as follows. Let \(Z^ 2_{pt}(A, C(\widehat{Z},\mathbb{T}))\) denote the subgroup of \(Z^ 2(A, C(\widehat{Z},\mathbb{T}))\) consisting of cocycles \(\omega\) which are pointwise trivial, that is, \(\omega_ \lambda \in B^ 2(A,\mathbb{T})\) for all \(\lambda \in \widehat{Z}\). Now suppose that \(A\) is discrete and that there exists \(\omega \in Z^ 2_{pt} (A, C(\widehat{Z}, \mathbb{T}))\) such that \(\sigma_ \lambda \omega_ \lambda| S_ \lambda \times S_ \lambda \equiv 1\) for all \(\lambda \in \widehat{Z}\). Then there exist a principal \(\widehat{A}\) bundle \((E_ \omega, \Phi, \widehat{Z})\) over \(\widehat{Z}\) and an action of \(N\) on \(E_ \omega\) such that \(\text{Prim}(C^*(N))\) is homeomorphic to the quasi-orbit space of the transformation group \((E_ \omega, N)\). Actually, this space \(E_ \omega\) has previously been studied in detail by Smith, Rosenberg and Raeburn and Williams. Finally, the authors apply their methods to compute the primitive ideal spaces of all two-step nilpotent, finitely generated, torsion-free discrete groups of rank less than or equal to five.