Let G be a discrete amenable group.

**General Question:** Let $J$ be an ideal of $C^*(G)$, the group C*-algebra of $G.$ Does $C^*(G)/J$ satisfy the universal
coefficient theorem (UCT)?

I am mainly interested in the following special case:

**Specific Question:** Same as the above question, but suppose moreover that G is finitely generated, nilpotent and $J$ is
the kernel of an irreducible
representation.

I know embarrassingly little about the UCT. I'm aware of Tu's 1999 paper "La conjecture de Baum-Connes pour les feuilletages moyennables" from which (I'm told) it follows that amenable groupoid C*-algebras satisfy the UCT, the various constructions that preserve the UCT ($\mathbb{Z}$-actions, inductive limits, two out of three property), and little else.

EDIT: Elizabeth Gillaspy and I recently showed the answer to the specific question is "yes" arXiv:1510.05469 [math.OA]. I still don't have any idea for the general case.

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