The $C^*$-algebras of completely solvable Lie groups are solvable
Ingrid Beltita, Daniel Beltita
TL;DR
The paper proves that the group $C^*$-algebra $C^*(G)$ of a completely solvable Lie group $G$ is solvable, meaning it admits a finite chain of closed two-sided ideals with quotients $C_0(\Gamma_j,\mathcal K(\mathcal H_j))$. The authors develop a framework combining continuous selections of polarizations with ultrafine layering to construct continuous fields of Hilbert spaces and controlled subquotients, extending prior nilpotent results. In the exponential/ completely solvable setting, they establish conditions under which continuous-trace subquotients are Morita equivalent to commutative algebras, and they show that each layer yields a trivial field, enabling a step-by-step decomposition of $C^*(G)$. The results provide a structural bridge between the Lie-algebraic solvability and the operator-algebraic solvability, with implications for understanding the spectrum and representation theory of solvable Lie groups. The nilpotent case aligns with earlier work (BBL16), while the present approach generalizes the scope using continuous-field and polarization techniques. Overall, the paper delivers a concrete, layer-by-layer decomposition of $C^*(G)$ for completely solvable $G$, highlighting the role of the Kirillov/Bernat correspondence and continuous-trace theory in the analysis of solvable group $C^*$-algebras.
Abstract
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 $\mathcal{J}_j/\mathcal{J}_{j-1}\simeq \mathcal{C}_0(Γ_j,\mathcal{K}(\mathcal{H}_j))$ for a suitable locally compact Hausdorff space $Γ_j$ and a separable complex Hilbert space $\mathcal{H}_j$, where $\mathcal{C}_0(Γ_j,\cdot)$ denotes the continuous mappings on $Γ_j$ that vanish at infinity, and $\mathcal{K}(\mathcal{H}_j)$ is the $C^*$-algebra of compact operators on $\mathcal{H}_j$ for $j=1,\dots,n$.