The $C^{\ast }$-algebras of completely solvable Lie groups are solvable

We prove that if a connected and simply connected Lie group $G$ admits connected closed normal subgroups $G_1\subseteq G_2\subseteq \cdots \subseteq G_m=G$ with $\dim G_j=j$ for $j=1,\dots ,m$, then its group $C^{\ast }$-algebra has closed two-sided ideals $\{0\}={\mathcal{J}}_0\subseteq {\mathcal{J}}_1\subseteq \cdots \subseteq {\mathcal{J}}_n=C^{\ast }(G)$ with ${\mathcal{J}}_j\mathrm{/}{\mathcal{J}}_{j-1}\simeq {\mathcal{C}}_0({\mathrm{\Gamma }}_j,\mathcal{K}({\mathcal{H}}_j))$ for a suitable locally compact Hausdorff space ${\mathrm{\Gamma }}_j$ and a separable complex Hilbert space ${\mathcal{H}}_j$, where ${\mathcal{C}}_0({\mathrm{\Gamma }}_j,\cdot )$ denotes the continuous mappings on ${\mathrm{\Gamma }}_j$ that vanish at infinity, and $\mathcal{K}({\mathcal{H}}_j)$ is the $C^{\ast }$-algebra of compact operators on ${\mathcal{H}}_j$ for $j=1,\dots ,n$.