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 $C^*$-algebra has closed two-sided ideals $\{ 0\}=\mathcal{ J}_0\subseteq \mathcal{ J}_1\subseteq\cdots\subseteq\mathcal{ J}_n=C^*(G)$ with $\mathcal{ J}_j/\mathcal{ J}_{ j-1}\simeq \mathcal{ C}_0(\Gamma_j,\mathcal{ K}(\mathcal{ H}_j))$ for a suitable locally compact Hausdorff space $\Gamma_j$ and a separable complex Hilbert space $\mathcal{ H}_j$, where $\mathcal{ C}_0(\Gamma_j,\cdot)$ denotes the continuous mappings on $\Gamma_j$ that vanish at infinity, and $\mathcal{ K}(\mathcal{ H}_j)$ is the $C^*$-algebra of compact operators on $\mathcal{ H}_j$ for $j=1,\dots,n$.
Categories: math.OA math.RT