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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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  • $\begingroup$ Did you ever find an answer to this question? $\endgroup$
    – Yemon Choi
    Commented May 18, 2015 at 20:25
  • $\begingroup$ @YemonChoi No, I just found several "I'm sure that's true"s from experts $\endgroup$
    – Caleb Eckhardt
    Commented May 19, 2015 at 12:49
  • $\begingroup$ This might be a difficult question because even taking $J$ to be trivial the question is whether the rather arbitrary amenable $C^*$-Algebra $C^*(G)$ satisfies UCT. But it is unsettled if every amenable (separable) $C^*$-Algebra is in the boot strap class. $\endgroup$
    – hänsel
    Commented Mar 20, 2016 at 22:23
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    $\begingroup$ @hansel The case you mention was settled by Tu in the paper I quoted in the original post. $\endgroup$
    – Caleb Eckhardt
    Commented Mar 27, 2016 at 3:45
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    $\begingroup$ @hansel Quotients of amenable C*-algebras are amenable. $\endgroup$
    – Caleb Eckhardt
    Commented Mar 30, 2016 at 14:10

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