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 $j=1,\dots,n$.
@article{JOLT_2025_35_4_a2,
author = {I. Beltita and D. Beltita},
title = {The {C*-Algebras} of {Completely} {Solvable} {Lie} {Groups} are {Solvable
}},
journal = {Journal of Lie Theory},
pages = {719--736},
year = {2025},
volume = {35},
number = {4},
doi = {10.5802/jolt.1406},
zbl = {08124769},
language = {en},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1406/}
}
TY - JOUR AU - I. Beltita AU - D. Beltita TI - The C*-Algebras of Completely Solvable Lie Groups are Solvable JO - Journal of Lie Theory PY - 2025 SP - 719 EP - 736 VL - 35 IS - 4 UR - https://jolt.centre-mersenne.org/articles/10.5802/jolt.1406/ DO - 10.5802/jolt.1406 LA - en ID - JOLT_2025_35_4_a2 ER -
I. Beltita; D. Beltita. The C*-Algebras of Completely Solvable Lie Groups are Solvable. Journal of Lie Theory, Volume 35 (2025) no. 4, pp. 719-736. doi: 10.5802/jolt.1406
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