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$C^*$-algebras generated by projective representations of free nilpotent groups

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 $C^*$-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 $C^*$-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 $C^*$-algebras associated with the free nilpotent groups of class $2$ and rank $n$ may be thought of as "second order" noncommutative $n$-tori.