Minimal \(H_3\) actions and simple quotients of discrete 7-dimensional nilpotent groups (Q1872580)

The paper under review extends earlier work by the author and S. Walters concerning simple quotients of the group \(C^*\)-algebras of cocompact discrete subgroups of nilpotent Lie groups of dimensions 4 to 6. The author discusses the simply connected nilpotent groups \(G_{7,a}\), \(G_{7,\beta}\) and \(G_{7,*}\) (where \(0<a\leq 1\) and \(\beta>0\)) corresponding to an infinite family of 7-dimensional nilpotent Lie algebras described by T. Skjelbred and T. Sund. The groups are shown to be pairwise non-isomorphic, and are characterized as the 7-dimensional central extensions of \(G_{6,15}\), with one-dimensional centre \({\mathbb{R}}\) (Theorem~1). For rational \(a\), the author defines a cocompact discrete subgroup \(H_{7,a}\) of \(G_{7,a}\). Depending on an irrational parameter \(\theta\), three unitarily inequivalent, irreducible representations of \(H_{7,a}\) are described (Theorem~3). For fixed \(\theta\), each of the representations generates, up to isomorphism, the same \(C^*\)-algebra \(A^{7,a}_\theta\), which is simple, has a unique tracial state, and can also be described either by generators and relations, or as a \(C^*\)-crossed product \(C^*({\mathcal C}({\mathbb{T}}^3), H_3)\) arising from an effective, minimal, distal flow of the discrete Heisenberg group \(H_3\) on the 3-dimensional torus (Theorem~2). Similar results are obtained for \(G_{7,\beta}\) and \(G_{7,*}\).