The Dixmier–Douady vanishing conjecture for subquotients of exponential Lie group $C^*$-algebras

Let $G$ be an exponential Lie group and let $C^*(G)$ be its $C^*$-algebra. A continuous-trace subquotient is a quotient $\mathcal J_2/\mathcal J_1$ of $C^*(G)$ by ideals $\mathcal J_1\subseteq\mathcal J_2$ that has continuous trace. Its Dixmier–Douady invariant is the obstruction to being Morita equivalent to a commutative $C^*$-algebra. Dixmier–Douady vanishing conjecture. Every continuous-trace subquotient of $C^*(G)$ has Dixmier–Douady invariant equal to zero. This conjecture is known for $2$-step nilpotent Lie groups, as well as for the additional families treated in the paper, but remains open for arbitrary exponential Lie groups.