Let G be an amenable group. I?ll discuss how an understanding of the extreme traces on G can make showing the UCT for C*-algebras generated by representations of G transparent. In fact in the cases I?ll discuss-G is virtually nilpotent-it will be so transparent that only the most basic facts about the UCT will be used. We?ll use this to show that C*-algebras generated by irreducible representations of finitely generated virtually nilpotent groups are classifiable and why this class of groups is special. I?ll end with plenty of open questions.