The Glimm ideal space of a two-step nilpotent locally compact group (Q2758145)

This paper studies the Glimm ideal space of the group \(C^*\)-algebra \(C^*(G) \) of a two-step nilpotent locally compact group \(G\). It is recalled that for a \(C^*\)-algebra \(A\), the Glimm ideal space \(\text{Glimm}(A) \) can be identified as the (topological) quotient space \(\text{Prim}(A) /\sim \) of Glimm classes, where \(\text{Prim}(A) \) is the primitive ideal space, consisting of kernels of irreducible *-representations of \(A\), endowed with the well-known Jacobson topology, and \(P\sim Q\) in \(\text{Prim}(A) \) if and only if \(P,Q\in \text{Prim}(A) \) cannot be separated by any (bounded) continuous function \(f\) on \(\text{Prim}(A) \), i.e. \(f(P) =f(Q) \). Indeed a Glimm class \([ P] \in \text{Prim}(A) /\sim \) corresponds to \(\bigcap_{Q\in [ P] }Q\in \text{Glimm}(A) \) under this identification. The authors first derive a description of the topological space \(\text{Glimm}(C^*(G)) \) for two-step nilpotent locally compact groups \(G\) by an explicit homeomorphic parametrization \( \mathcal{G}\to \text{Glimm}(C^*(G)) \) sending \((\lambda ,F_{\lambda },\tau) \) to \(\ker (\operatorname {ind}_{F_{\lambda }}^{G}\tau) \), where \(\mathcal{G}\) is a topological space consisting of triple \((\lambda ,F_{\lambda },\tau) \) with \(\lambda \in \widehat{Z}\) for the center \(Z\) of \(G\) and a \(G\)-invariant character \(\tau \) of a closed subgroup \(F_{\lambda }\) of \(G\) associated to \( \lambda \) and containing \(Z\) such that \(\tau |_{Z}=\lambda \). Then they establish that the quotient map from \(\text{Prim}(C^*(G)) \) onto \(\text{Glimm}(C^*(G)) \) is open if and only if the map \(\lambda \mapsto F_{\lambda }\) from \(\widehat{Z}\) to \( \mathcal{K}(G) \) is continuous, where the set \(\mathcal{K}(G) \) of closed subgroups of \(G\) is endowed with the compact-open topology. As a consequence, they obtain a necessary and sufficient condition for the \(C^*\)-algebra \(C^*(G) \) to be quasi-standard. Finally, Glimm classes with special conditions, e.g. \([ \ker (1_{G}) ] \) with \(\overline{[ G,G] }\) having a maximal compact subgroup, are determined and some concrete examples of Glimm ideal spaces are computed.