$C^{\ast }$-algebras generated by projective representations of free nilpotent groups

Abstract

We compute the two-cocycles (or multipliers) of the free nilpotent groups of class $2$ and rank $n$ and give conditions for simplicity of the corresponding twisted group $C^{\ast }$-algebras. These groups are representation groups for ${\mathbb{Z}}^n$ and can be considered as a family of generalized Heisenberg groups with higher-dimensional center. Their group $C^{\ast }$-algebras are in a natural way isomorphic to continuous fields over ${\mathbb{T}}^{\frac{1}{2}n(n-1)}$ with the noncommutative $n$-tori as fibers. In this way, the twisted group $C^{\ast }$-algebras associated with the free nilpotent groups of class $2$ and rank $n$ may be thought of as "second order" noncommutative $n$-tori.

24 pages; this version includes some generalizations and other details that were left out in the published version

References in corpus (2)

• Primeness and primitivity conditions for twisted group $C^{\ast }$-algebras
• Twisted homological stability for extensions and automorphism groups of free nilpotent groups

Cited by in corpus (3)

• Primeness and primitivity conditions for twisted group $C^{\ast }$-algebras
• Smooth lattice orbits of nilpotent groups and strict comparison of projections
• Free nilpotent groups are C*-superrigid