The $C^*$-algebras of completely solvable Lie groups are solvable

TL;DR

This paper proves that the $C^*$-algebras of certain connected, simply connected, completely solvable Lie groups have a specific ideal structure, decomposing into continuous functions vanishing at infinity with compact operator fibers.

Contribution

It establishes a detailed ideal decomposition for the $C^*$-algebras of completely solvable Lie groups with a specific normal subgroup chain.

Findings

The $C^*$-algebra has a finite chain of closed two-sided ideals.

Each quotient in the chain is isomorphic to continuous functions vanishing at infinity with compact operator fibers.

The structure provides insight into the representation theory of these Lie groups.

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 $\mathcal{J}_j/\mathcal{J}_{j-1}\simeq \mathcal{C}_0(\Gamma_j,\mathcal{K}(\mathcal{H}_j))$ 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 $j=1,\dots,n$.