It was recently shown that each C*-algebra generated by a faithful irreducible representation of a finitely generated, torsion free nilpotent group is classified by its ordered K-theory. For the three step nilpotent group $UT(4,\mathbb{ Z})$ we calculate the ordered K-theory of each C*-algebra generated by a faithful irreducible representation of $UT(4,\mathbb{ Z})$ and see that they are all simple A$\mathbb{ T}$ algebras. We also point out that there are many simple non A$\mathbb{ T}$ algebras generated by irreducible representations of nilpotent groups.
Categories: math.OA math.KT