Fourier transforms of $C^*$-algebras of nilpotent Lie groups

For any nilpotent Lie group $G$ we provide a description of the image of its $C^*$-algebra through its operator-valued Fourier transform. Specifically, we show that $C^*(G)$ admits a finite composition series such that that the spectra of the corresponding quotients are Hausdorff sets in the relative topology, defined in terms of the fine stratification of the space of coadjoint orbits of $G$, and the canonical fields of elementary $C^*$-algebras defined by the successive subquotients are trivial. We give a description of the image of the Fourier transform as a $C^*$-algebra of piecewise continuous operator fields on the spectrum, determined by the boundary behavior of the restrictions of operator fields to the spectra of the subquotients in the composition series. For uncountable families of 3-step nilpotent Lie groups and also for a sequence of nilpotent Lie groups of arbitrarily high nilpotency step, we prove that every continuous trace subquotient of their $C^*$-algebras has its Dixmier-Douady invariant equal to zero.