Journal of Lie Theory 35 (2025), No. 4, 719--736
Copyright Heldermann Verlag 2025
The C*-Algebras of Completely Solvable Lie Groups are Solvable
Ingrid Beltita
Institute of Mathematics "Simion Stoilow", Romanian Academy, Bucharest, Romania
ingrid.beltita@imar.ro
Daniel Beltita
Institute of Mathematics "Simion Stoilow", Romanian Academy, Bucharest, Romania
daniel.beltita@imar.ro
[Abstract-pdf]
\newcommand{\Cc}{{\mathcal C}} \newcommand{\Hc}{{\mathcal H}} \newcommand{\Jc}{{\mathcal J}} \newcommand{\Kc}{{\mathcal K}} 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 $\{0\}=\Jc_0\subseteq \Jc_1\subseteq\cdots\subseteq\Jc_n=C^*(G)$ with $\Jc_j/\Jc_{j-1}\simeq \Cc_0(\Gamma_j,\Kc(\Hc_j))$ for a suitable locally compact Hausdorff space $\Gamma_j$ and a separable complex Hilbert space $\Hc_j$, where $\Cc_0(\Gamma_j,\cdot)$ denotes the continuous mappings on $\Gamma_j$ that vanish at infinity, and $\Kc(\Hc_j)$ is the $C^*$-algebra of compact operators on $\Hc_j$ for $j=1,\dots,n$.
Keywords: Completely solvable Lie group, solvable C*-algebra.
MSC: 22E27; 17B30, 46L05, 46L55.
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