Structure of group \(C^*\)-algebras of the generalized Mautner groups. (Q1411730)

In this paper, a generalized Mautner group \(M_{n,m}\) is, by definition, a semi-direct product \(\mathbb{C}^{n}\rtimes_{\alpha}\mathbb{R}^{m}\) of the abelian Lie groups \(\mathbb{C}^{n}\) and \(\mathbb{R}^{m}\) with the action \(\alpha\) of \(\mathbb{R}^{m}\) on \(\mathbb{C}^{n}\) defined by diagonal matrices \(\alpha_{t}=\text{diag}\left( z_{1},\dots,z_{n}\right) \) for \(t\in\mathbb{R}^{m}\) where \(z_{j}=\exp\left( 2\pi i\sum_{k=1}^{m}c_{jk} t_{k}\right) \) for some constants \(c_{jk}\in\mathbb{R}\). Such a group is simply connected solvable, and \(M_{2,1}\) with \(c_{11}=1\) and \(c_{21}\) irrational is a Mautner group. The author shows that the (full) group C*-algebra \(C^{\ast}\left( M_{n,m}\right) \) has a finite composition series \(\left\{ \mathfrak{I}_{j}\right\} \) with subquotients \(\mathfrak{I}_{j} /\mathfrak{I}_{j-1}\) being tensor products of commutative C*-algebras, noncommutative tori, and the algebra of compact operators. It is also shown that \(C^{\ast}\left( M_{n,m}\right) \) for \(n,m\geq2\) contains no nontrivial projections. Furthermore the stable rank and the connected stable rank of \(C^{\ast}\left( M_{n,m}\right) \) are estimated.