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Journal of Operator Theory

Volume 73, Issue 1, Winter 2015  pp. 3-25.

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

Authors:  Tron Anen Omland
Author institution:Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway

Summary: We compute the multipliers (two-cocycles) 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.

Keywords: Free nilpotent group, projective unitary representation, twisted group $C^*$-algebra, simplicity, multiplier, two-cocycle, group cohomology, Heisenberg group, noncommutative $n$-torus.

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