The ideal structure of C*-algebras of etale groupoids with isotropy groups of local polynomial growth
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911056744415232 |
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| author | Christensen, Johannes Neshveyev, Sergey |
| author_facet | Christensen, Johannes Neshveyev, Sergey |
| contents | Given an amenable second countable Hausdorff locally compact étale groupoid $\mathcal G$ such that each isotropy group $\mathcal G^x_x$ has local polynomial growth, we give a description of $\operatorname{Prim} C^*(\mathcal G)$ as a topological space in terms of the topology on $\mathcal G$ and representation theory of the isotropy groups and their subgroups. The description simplifies when either the isotropy groups are FC-hypercentral or $\mathcal G$ is the transformation groupoid $Γ\ltimes X$ defined by an action $Γ\curvearrowright X$ with locally finite stabilizers. To illustrate the class of C$^*$-algebras for which our results can provide a complete description of the ideal structure, we compute the primitive spectrum of $\mathrm{SL}_3(\mathbb Z)\ltimes C_0(\mathrm{SL}_3(\mathbb R)/U_3(\mathbb R))$, where $U_3(\mathbb R)$ is the group of unipotent upper triangular matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11805 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The ideal structure of C*-algebras of etale groupoids with isotropy groups of local polynomial growth Christensen, Johannes Neshveyev, Sergey Operator Algebras Representation Theory Given an amenable second countable Hausdorff locally compact étale groupoid $\mathcal G$ such that each isotropy group $\mathcal G^x_x$ has local polynomial growth, we give a description of $\operatorname{Prim} C^*(\mathcal G)$ as a topological space in terms of the topology on $\mathcal G$ and representation theory of the isotropy groups and their subgroups. The description simplifies when either the isotropy groups are FC-hypercentral or $\mathcal G$ is the transformation groupoid $Γ\ltimes X$ defined by an action $Γ\curvearrowright X$ with locally finite stabilizers. To illustrate the class of C$^*$-algebras for which our results can provide a complete description of the ideal structure, we compute the primitive spectrum of $\mathrm{SL}_3(\mathbb Z)\ltimes C_0(\mathrm{SL}_3(\mathbb R)/U_3(\mathbb R))$, where $U_3(\mathbb R)$ is the group of unipotent upper triangular matrices. |
| title | The ideal structure of C*-algebras of etale groupoids with isotropy groups of local polynomial growth |
| topic | Operator Algebras Representation Theory |
| url | https://arxiv.org/abs/2412.11805 |