The \(C^{*}\)-algebra of some 6-dimensional nilpotent Lie groups (Q2877574)

This paper presents a detailed case-by-case study of the unitary duals and \(C^*\)-algebras of four 6-dimensional nilpotent Lie groups (namely, the groups \(G_{6,4}\), \(G_{6,5}\), \(G_{6,6}\) and \(G_{6,8}\) in the notation of [\textit{O. A. Nielsen}, Unitary representations and coadjoint orbits of low-dimensional nilpotent Lie groups. Ontario, Canada: Queen's University (1983; Zbl 0591.22004)]). The unitary duals, and explicit models for the irreducible representations, are computed using Kirillov's orbit method (see [\textit{A. A. Kirillov}, Russ. Math. Surv. 17, No. 4, 53--104 (1962); translation from Usp. Mat. Nauk 17, No. 4(106), 57--110 (1962; Zbl 0106.25001)] and [\textit{I. D. Brown}, Ann. Sci. Éc. Norm. Supér. (4) 6, 407--411 (1973; Zbl 0284.57026)]). These computations are then used to prove the main result of the paper: for each of the groups \(G\) under consideration, the group \(C^*\)-algebra \(C^*(G)\) has the property of \textit{norm-controlled dual limits} introduced by the authors in their earlier study of 5-dimensional nilpotent groups [\textit{J. Ludwig} and \textit{H. Regeiba}, ``\(C^{*}\) -algebras with norm controlled dual limits and nilpotent Lie groups'', Preprint, \url{arXiv:1309.6941}]. As the authors demonstrated in [op. cit.], this property allows one to describe \(C^*(G)\) explicitly as an algebra of compact-operator-valued functions on the unitary dual \(\widehat{G}\).