The $C^*$-algebras of completely solvable Lie groups are solvable
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915239317995520 |
|---|---|
| author | Beltita, Ingrid Beltita, Daniel |
| author_facet | Beltita, Ingrid Beltita, Daniel |
| contents | 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 $j=1,\dots,n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13923 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $C^*$-algebras of completely solvable Lie groups are solvable Beltita, Ingrid Beltita, Daniel Operator Algebras Representation Theory Primary 22E27, Secondary 17B30, 46L05, 46L55 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 $j=1,\dots,n$. |
| title | The $C^*$-algebras of completely solvable Lie groups are solvable |
| topic | Operator Algebras Representation Theory Primary 22E27, Secondary 17B30, 46L05, 46L55 |
| url | https://arxiv.org/abs/2412.13923 |