# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2013mar06.2037

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

Contents
Full-Text PDF